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Matrix Formulae and Skein Relations for Quasi-Cluster Algebras

Cody Gilbert Saint Louis University cody.gilbert@slu.edu McCleary Philbin University of Wisconsin - River Falls, River Falls, WI mccleary.philbin@uwrf.edu  and  Kayla Wright University of Minnesota, Twin Cities, Minneapolis, MN kaylaw@umn.edu
Abstract.

In this paper, we give matrix formulae for non-orientable surfaces that provide the Laurent expansion for quasi-cluster variables, generalizing the orientable surface matrix formulae by Musiker-Williams. We additionally use our matrix formulas to prove the skein relations for the elements in the quasi-cluster algebra associated to curves on the non-orientable surface.

1. Introduction

Cluster algebras were first defined by Fomin and Zelevinksy in the early 2000’s in an effort to study problems regarding dual canonical bases and total positivity [5]. Since their axiomatization, endowing mathematical objects with a cluster structure has become a rapidly growing field of interest across many different disciplines in math and physics. For instance, Fomin and Shapiro defined a cluster structure associated to orientable topological surfaces (which actually generalized the deep geometric work on Fock and Goncharov in [2, 3] and Gekhtman, Shapiro, and Vainshtein [6]). The cluster algebra structure associated to an orientable surface has a completely topological description, but also can be bolstered to include more geometric significance. Namely, the cluster structure also arises from coordinate rings for the decorated Teichml̈ler space associated to the surface. The cluster variables can be thought of as hyperbolic lengths of geodesics on the surfaces also known as Penner coordinates [9]. Because of this interpretation, the Teichmüller theory was connected to combinatorics by associating elements in PSL2()PSL_{2}(\mathbb{R}) to arcs on the surface in order to obtain the Laurent expansions of the associated cluster variables. This result was first studied by Fock and Goncharov in [2, 3] in the coefficient-free case and generalized by Musiker and Williams in [8].

More recently, cluster-like structures were defined for non-orientable surfaces by Dupont and Palesi [1]. They defined quasi-cluster algebras associated to non-orientable surfaces drawing inspiration from the orientable case in [4]. The way they define their mutation or exchange relations is also inspired from the Teichmüller theory and geometry from [2, 3]. Positivity for quasi-cluster algebras was recently proven by Wilson through using snake and band graph combinatorics in [11] inspired by the original proof for positivity for cluster algebras from orientable surfaces in [7].

In our paper, we create explicit matrix formulae for non-orientable surfaces that give the Laurent expansions for quasi-cluster variables in the associated quasi-cluster algebra. That is, we associate a product of matrices in PSL2()PSL_{2}(\mathbb{R}) to a curve on a non-orientable surface so that the Laurent expansion for the associated element of the quasi-cluster algebra can be extracted from the matrix. Our method provides the first non-recursive way to compute Laurent expansions for quasi-cluster variables directly on the non-orientable surface. Our process modifies the construction by Musiker and Williams [8] by associated an element of PSL2()PSL_{2}(\mathbb{R}) when traversing through the non-trivial topology of a non-orientable surface. We in turn prove our result by relying on the combinatorics of snake and band graphs and a coefficient system using the language of laminations and Shear coordinates for non-orientable surfaces developed by Wilson in [11]. We also use our results to prove the skein relations for non-orientable surfaces.

Our paper is organized as follows: in Section 2 we review the cluster-like structure of non-orientable surfaces and topological notions we use throughout the paper. Section 3 introduces the combinatorics of snake and band graphs in order to state the expansion formula for quasi-cluster variables as in [11]. Section 4 introduces the coefficient system via laminations that give arbitrary coefficients for quasi-cluster algebras. Section 5 recalls the matrix formulae by Musiker and Williams for cluster algebras from orientable surfaces and also extends their results to the context of prinicipal laminations. This paves the way to our main results Theorem 6.4 and Theorem 6.8 that is then proven in Section 6. Finally, Section 7 proves the skein relations for non-orientable surfaces.

Acknowledgements. The authors would like to thank Chris Fraser for suggesting this project and helping us with the beginning stages of learning the relevant background. We would also like to thank Gregg Musiker for helpful comments and discussions.

2. Quasi-Cluster Algebras

In this section, we review the definition of quasi-cluster algebras defined in [1]. We give explicit computations for the so-called quasi-mutation rules that come from skein relations as this was missing from the literature.

Definition 2.1.

Let 𝐒\mathbf{S} be a compact, connected Riemann surface with boundary 𝐒\partial\mathbf{S}. Let 𝐌\mathbf{M} be a finite set of points, we call marked points, contained in 𝐒\mathbf{S} such that each connected component of 𝐒\partial\mathbf{S} has at least one point of 𝐌\mathbf{M}. We say the pair (𝐒,𝐌)(\mathbf{S},\mathbf{M}) is a marked surface.

We call marked points on the interior of 𝐒\mathbf{S} punctures.

Definition 2.2.

A regular arc τ\tau in a marked surface (𝐒,𝐌)(\mathbf{S},\mathbf{M}) is a curve in 𝐒\mathbf{S}, considered up to isotopy relative its endpoints such that

  1. (1)

    the endpoints of τ\tau are 𝐌\mathbf{M},

  2. (2)

    τ\tau has no self-intersections, except possibly at its endpoints,

  3. (3)

    except for its endpoints, τ\tau does not intersect 𝐌𝐒\mathbf{M}\cup\partial\mathbf{S};

  4. (4)

    and τ\tau does not cut out a monogon or bigon.

We say the arc is a generalized arc if τ\tau intersects itself, dropping condition (2). We say the arcs that connect two marked points and lie completely on 𝐒\partial\mathbf{S} are boundary arcs.

In our paper, we will be analyzing the specific case when 𝐒\mathbf{S} is non-orientable. By the classification of compact surface, any non-orientable surface is homeomorphic to the connected sum of kk projective planes. By this fact, we refer to kk as the non-orientable genus of the surface. Recalling that the projective plane is a topological quotient of the 2-sphere by the antipodal map, we visualize these surfaces with crosscaps \bigotimes. This symbol denotes the removal of a closed disk with the antipodal points identified.

On a non-orientable surface 𝐒\mathbf{S}, a closed curve is said to be two-sided if it admits an orientable regular neighborhood. It a closed curve does not admit such a neighborhood, it is said to be one-sided. Since one-sided curves reverse the local orientation, they may only be contained in a non-orientable surface.

As in the orientable case, an arc is an isotopy class of a simple curve in (𝐒,𝐌)(\mathbf{S},\mathbf{M}).

Definition 2.3.

A quasi-arc in (𝐒,𝐌)(\mathbf{S},\mathbf{M}) is either an arc or a simple one-sided closed curve in the interior of 𝐒\mathbf{S}.

We say that two arcs τ,τ\tau,\tau^{\prime} in (𝐒,𝐌)(\mathbf{S},\mathbf{M}) are compatible if up to isotopy, they do not intersect one another. More formally, define e(τ,τ)e(\tau,\tau^{\prime}) to be the minimum of the number of crossings of α\alpha and α\alpha’ where α\alpha is an arc isotopic to τ\tau and α\alpha^{\prime} is an arc isotopic to τ\tau^{\prime}

e(τ,τ):=min{#of crossings of α and α|ατ and ατ}e(\tau,\tau^{\prime}):=\min\{\#\text{of crossings of }\alpha\text{ and }\alpha^{\prime}\leavevmode\nobreak\ |\leavevmode\nobreak\ \alpha\simeq\tau\text{ and }\alpha^{\prime}\simeq\tau^{\prime}\}

where α\alpha ranges over arcs that are isotopic to τ\tau and α\alpha^{\prime} ranges over arcs isotopic to τ\tau^{\prime}.

We say regular arcs τ,τ\tau,\tau^{\prime} are compatible if (eτ,τ)=0(e\tau,\tau^{\prime})=0. A maximal collection of pairwise compatible arcs in (𝐒,𝐌)(\mathbf{S},\mathbf{M}) is called a quasi-triangulation of (𝐒,𝐌)(\mathbf{S},\mathbf{M}). If none of the arcs in this collection is a quasi-arc, we say it is simply a triangulation.

All (quasi-)triangulations of a non-orientable surface are reachable via sequences of quasi-mutations [1, Proposition 3.12]. These quasi-mutations are a larger class of local moves on the surface that are motivated by the skein relations, studied in cluster algebra theory by [4, 8]. To discuss mutations we require the notion of a quasi-seed.

Let nn be the rank and bb the number of boundary components of the marked surface (𝐒,𝐌)(\mathbf{S},\mathbf{M}) and \mathcal{F} a field of rational functions in n+bn+b indeterminates. For each boundary component aa, we associate a variable xax_{a}\in\mathcal{F} such that {xa|a𝐒}\{x_{a}|a\in\partial\mathbf{S}\} is algebraically independent in \mathcal{F}. We refer to =[xa±|a𝐒}\mathbb{Z}\mathbb{P}=\mathbb{Z}[x_{a}^{\pm}|a\in\partial\mathbf{S}\}\subset\mathcal{F} as the ground ring.

Definition 2.4.

A quasi-seed associated with (𝐒,𝐌)(\mathbf{S},\mathbf{M}) in \mathcal{F} is a pair Σ=(T,𝐱)\Sigma=(T,\mathbf{x}) such that

  • TT is a quasi-triangulation of (𝐒,𝐌)(\mathbf{S},\mathbf{M});

  • 𝐱={xt|tT}\mathbf{x}=\{x_{t}\,|\,t\in T\} is a free generating set of the field \mathcal{F} over \mathbb{Z}\mathbb{P}.

The set {xt|tT}\{x_{t}\,|\,t\in T\} is called the quasi-cluster of the quasi-seed Σ\Sigma. We say a quasi-seed is a seed if the corresponding quasi-triangulation is a triangulation.

Definition 2.5.

An anti-self-folded triangle is any triangle of a quasi-triangulation with two edges identified by an orientation-reversing isometry.

For example, in Figure 1, the triangle with sides (t,t,a)(t,t,a) is an anti-self-folded triangle.

Before stating the quasi-mutation rules, we derive two of the mutations from the skein relations. The other, significantly longer, computation is given in the appendix.

Example 2.6.

Let tt be an arc in the anti-self-folded triangles with sides (t,t,a)(t,t,a), and let tt^{\prime} be a one-sided curve in an annuli with boundary aa. Figure 1 demonstrates how to resolve the crossing of tt and tt^{\prime} using the Ptolemy relation, which yields the relation xtxt=xax_{t}x_{t^{\prime}}=x_{a}. We explain how to push a loop through the crosscap in the appendix. With this we can derive the second and third quasi-mutations given in Definition 2.7.

Refer to caption
Figure 1. The skein relations used to define quasi-mutation (3).

The justification for pushing the loop through the crosscap is given in the appendix.

With this motivation, we have the following definitions of quasi-mutation.

Definition 2.7.

Given tTt\in T, the quasi-mutation of Σ\Sigma in the direction TT as the pair μt(T,𝐱)=(T,𝐱)\mu_{t}(T,\mathbf{x})=(T^{\prime},\mathbf{x^{\prime}}) where T=μt(T)=T{t}{t}T^{\prime}=\mu_{t}(T)=T\setminus\{t\}\sqcup\{t^{\prime}\} and 𝐱={xv|vT}\mathbf{x^{\prime}}=\{x_{v}\,|\,v\in T^{\prime}\} such that xtx_{t^{\prime}} is defined as follows:

  1. (1)

    If tt is an arc separating two distinct triangles with sides (a,b,t)(a,b,t) and (c,d,t)(c,d,t), then the relation is given by the Ptolemy relation for arcs xtxt=xaxc+xbxdx_{t}x_{t^{\prime}}=x_{a}x_{c}+x_{b}x_{d}.

    [Uncaptioned image]
  2. (2)

    If tt is an arc in an anti-self-folded triangle with sides (t,t,a)(t,t,a), then the relation is xtxt=xax_{t}x_{t^{\prime}}=x_{a}.

    [Uncaptioned image]
  3. (3)

    If tt is a one-sided curve in an annuli with boundary aa, then the relation is xtxt=xax_{t}x_{t^{\prime}}=x_{a}.

    [Uncaptioned image]
  4. (4)

    If tt is an arc separating a triangle with sides (a,b,t)(a,b,t) and an annuli with boundary tt and one-sided curve dd, then the relation is xtxt=(xa+xb)2+xd2xaxbx_{t}x_{t^{\prime}}=(x_{a}+x_{b})^{2}+x_{d}^{2}x_{a}x_{b}.

    [Uncaptioned image]

Now that we have the necessary topological notions, we are ready to define the cluster structure for these non-orientable surface. This was first defined in [1] which was inspired by work in [4] in the orientable setting.

Definition 2.8.

Let 𝒳\mathcal{X} be the collection of all quasi-cluster variables obtained by iterated quasi-mutation from an initial seed Σ\Sigma. The quasi-cluster algebra is the polynomial ring generated by the quasi-cluster variables over the ground ring \mathbb{Z}\mathbb{P} i.e. 𝒜(Σ)=[𝒳]\mathcal{A}(\Sigma)=\mathbb{Z}\mathbb{P}[\mathcal{X}].

3. Expansion Formula via Snake and Band Graphs for Non-Orientable Surfaces

In this section, we review the expansion formulae for quasi-cluster variables in terms of snake and band graphs [11] without coefficients. We postpone the discussion on coefficients until 4. We begin by defining the notion of a snake graph as in [7] and then review the definition of band graphs that come from non-orientable surfaces as in [11].

Definition 3.1.

A tile is a copy of the cycle graph C4C_{4} on four vertices, embedded in 2\mathbb{R}^{2} as a square with four cardinal directions, see Figure 2.

NNEESSWW
Figure 2. A tile with the four cardinal directions.

We glue tiles together in a particular way to obtain a snake or band graph. In particular, a snake or band graph can be thought of as a sequence of tiles glued along either the North or East edge of the previous tile. We describe how to construct a snake graph from an arc on a triangulated orientable surface.

Definition 3.2.

Let γ\gamma be an arc overlayed on a triangulation TT of an orientable surface (𝐒,𝐌)(\mathbf{S},\mathbf{M}). The snake graph associated to γ\gamma is a sequence of tiles 𝒢γ=(G1,,Gd)\mathcal{G}_{\gamma}=(G_{1},\dots,G_{d}) where

  • the tiles GiG_{i} correspond to the arc τi\tau_{i} of TT that γ\gamma intersects;

  • and we attach GiG_{i} to Gi+1G_{i+1} along the unique shared edge of the local quadrilaterals containing τi\tau_{i} and τi+1\tau_{i+1}.

Label the N, E, S, W edges of each tile with the corresponding arcs in the local quadrilateral so that the relative orientation of the tiles alternate, see [7, 11] for details.

We now give the definition of a band graph associated to a one-sided closed curve as in [11].

Definition 3.3.

Let γ\gamma be a one-sided closed curve overlayed on a quasi-triangulation TT, without quasi-arcs, of a non-orientable surface (𝐒,𝐌)(\mathbf{S},\mathbf{M}). Let xx be a point on γT\gamma\setminus T. Consider the orientable double cover (𝐒,𝐌)~\widetilde{(\mathbf{S},\mathbf{M})} of (𝐒,𝐌)(\mathbf{S},\mathbf{M}) along the the lifts of TT, γ\gamma and xx. Note that T~\tilde{T} is a triangulation of (𝐒,𝐌)~\widetilde{(\mathbf{S},\mathbf{M})}, γ~\tilde{\gamma} is an orientable closed curve and x1~\tilde{x_{1}}, x2~\tilde{x_{2}} of xx are antipodal points on γ~\tilde{\gamma}.

Let 𝒢γ,T,x=(G1,,Gd)\mathcal{G}_{\gamma,T,x}=(G_{1},\dots,G_{d}) be the snake graph of the arc corresponding to tracing along γ~\tilde{\gamma} from x1~\tilde{x_{1}} to x2~\tilde{x_{2}} clockwise. If we continue along γ~\tilde{\gamma} through one more intersection with T~\tilde{T}, we’d obtain (G1,,Gd,Gd+1)(G_{1},\dots,G_{d},G_{d+1}) where G1G_{1} and Gd+1G_{d+1} are lifts of the same local quadrilateral in TT. Let bb be the glued edge between GdG_{d} and Gd+1G_{d+1} and b¯\overline{b} be the corresponding edge in G1G_{1}. The band graph associated to γ\gamma, denoted 𝒢γ,x\mathcal{G}_{\gamma,x} is the snake graph 𝒢γ~=(G1,,Gd)\mathcal{G}_{\tilde{\gamma}}=(G_{1},\dots,G_{d}) with bb and b¯\overline{b} identified.

Remark 3.4.

Up to isomorphism, the choice of point xx on the one-sided closed curve γ\gamma does not affect the band graph or expansion formulae [11].

To state the expansion formulae, we need the notion of (good) perfect matchings on snake (band) graphs.

Definition 3.5.

A perfect matching of a snake graph 𝒢\mathcal{G} is a subset PP of the edges of GG that covers each vertex exactly once. A good perfect matching of a band graph 𝒢¯=(G1,,Gd)\overline{\mathcal{G}}=(G_{1},\dots,G_{d}), glued along the edge bb from x--yx\relbar\mkern-9.0mu\relbar y, is a perfect matching PP of the un-identified underlying snake graph 𝒢\mathcal{G} where xx and yy are either matched with two edges are on G1G_{1} or on GdG_{d}.

Now, we state the expansion formula for regular arcs without coefficients as in Theorem 5.19 in [11]:

Theorem 3.6.

[11] Let γ\gamma be a regular arc overlayed on a quasi-triangulation TT without quasi-arcs of a non-orientable (𝐒,𝐌)(\mathbf{S},\mathbf{M}). Let γ~\tilde{\gamma} be one of the two lifts of γ\gamma on the orientable double cover (𝐒,𝐌)(\mathbf{S},\mathbf{M}). Let 𝒢γ~\mathcal{G}_{\tilde{\gamma}} be the snake graph associated to γ~\tilde{\gamma} and 𝒫\mathcal{P} be the set of perfect matchings on 𝒢γ~\mathcal{G}_{\tilde{\gamma}}. Then the quasi-cluster variable xγx_{\gamma} can be expressed as follows:

xγ=1crossT(γ)P𝒫x(P)x_{\gamma}=\frac{1}{\text{cross}_{T}(\gamma)}\sum_{P\in\mathcal{P}}x(P)

where cross(γ)T=xi1xid{}_{T}(\gamma)=x_{i_{1}}\cdots x_{i_{d}} is the crossing monomial which keeps track of the arcs of the triangulation τi1,,τid\tau_{i_{1}},\dots,\tau_{i_{d}} that γ\gamma crosses, counting multiplicities, and x(P)x(P) is the product of all the edge labels of 𝒢γ~\mathcal{G}_{\tilde{\gamma}} in PP.

The analogous expansion formula for quasi-arcs without coefficients is given in Theorem of [11]:

Theorem 3.7.

[11] Let γ\gamma be a one-sided closed curve overlayed on a quasi-triangulation TT without quasi-arcs of a non-orientable (𝐒,𝐌)(\mathbf{S},\mathbf{M}). Let γ~\tilde{\gamma} be the lift of γ\gamma on the orientable double cover (𝐒,𝐌)(\mathbf{S},\mathbf{M}). Let 𝒢γ~\mathcal{G}_{\tilde{\gamma}} be the band graph associated to γ~\tilde{\gamma} and 𝒫\mathcal{P} be the set of good perfect matchings on 𝒢γ~\mathcal{G}_{\tilde{\gamma}}. Then the quasi-cluster variable xγx_{\gamma} can be expressed as follows:

xγ=1crossT(γ)P𝒫x(P)x_{\gamma}=\frac{1}{\text{cross}_{T}(\gamma)}\sum_{P\in\mathcal{P}}x(P)

where cross(γ)T=xi1xid{}_{T}(\gamma)=x_{i_{1}}\cdots x_{i_{d}} is the crossing monomial which keeps track of the arcs of the triangulation τi1,,τid\tau_{i_{1}},\dots,\tau_{i_{d}} that γ\gamma crosses, counting multiplicities, and x(P)x(P) is the product of all the edge labels of 𝒢γ~\mathcal{G}_{\tilde{\gamma}} in PP.

4. Coefficients Using Principal Laminations

We review the arbitrary coefficients defined in [11] via principal laminations that complete the expansion formula mentioned in Section 3. We will then take inspiration from this coefficient system to define a poset structure of the set of good perfect matchings associated to a quasi-arc γ\gamma in a marked surface (𝐒,𝐌)(\mathbf{S},\mathbf{M}).

Definition 4.1.

A set of self-non-intersecting and pairwise non-intersecting curves LL on a marked surface (𝐒,𝐌)(\mathbf{S},\mathbf{M}) is called a lamination if each L\ell\in L is any of the following curves:

  • a one-sided closed curve;

  • a two-sided closed curve that does not bound a disk, or a Möbius strip;

  • a curve that connected two points on 𝐒𝐌\partial\mathbf{S}\setminus\mathbf{M} that is not isotopic to a boundary arc.

A multilamination \mathbf{\mathcal{L}} of (𝐒,𝐌)(\mathbf{S},\mathbf{M}) is a finite collection of laminations of (𝐒,𝐌)(\mathbf{S},\mathbf{M}).

Example 4.2.

The figure below is an example of a multilamination in red on the Klein bottle with four marked points.

We now define a principal lamination for marked surfaces in the unpunctured case.

Definition 4.3.

Let γ\gamma be an arc in (𝐒,𝐌)(\mathbf{S},\mathbf{M}). We associate the lamination LγL_{\gamma} to the curve γ\gamma via the following:

  • if γ\gamma is a regular arc, take LγL_{\gamma} to be a lamination that runs along γ\gamma in a small neighborhood, but turns clockwise (counterclockwise) at the marked point and ends when at the boundary.

  • if γ\gamma is a quasi-arc, take LγL_{\gamma} to be either a lamination that runs along γ\gamma in a small neighborhood and has endpoints on the boundary or take LγL_{\gamma} to be the 1 sided closed curve that is compatible with γ\gamma.

γ\gammaγ\gammaLγL_{\gamma}LγL_{\gamma}aaaa^{\prime}bbbb^{\prime}
Figure 3. Examples of the conditions for a laminations. On the left, the case where γ\gamma is not a quasi-arc and on the right, the case where γ\gamma is a quasi-arc.
Definition 4.4.

Let TT be a triangulation of (𝐒,𝐌)(\mathbf{S},\mathbf{M}). Taking the collection of all laminations associated to the arcs τT\tau\in T is called a principal lamination. That is, a multilamination of the form T={Lτ:τT}\mathcal{L}_{T}=\{L_{\tau}\leavevmode\nobreak\ :\leavevmode\nobreak\ \tau\in T\} is a principal lamination.

In order to define the coefficients seen in [11], we must use the notion of a principal lamination to define Shear coordinates, a coordinate we place on the diagonal of a local quadrilateral in a triangulation.

γ{\displaystyle\gamma}γ{\displaystyle\gamma}γ{\displaystyle\gamma}LγL\gammaLγL\gammaClockwise to γ\gammaCounterclockwise to γ\gammaSS intersectionZZ intersectionbT(Lγ,γ)= 1b_{T}(L_{\gamma},\gamma)\ =\ 1bT(Lγ,γ)=1b_{T}(L_{\gamma},\gamma)\ =-1bT(L,γ)= 2b_{T}(L,\gamma)\ =\ 2LL
Figure 4. The definition of SS and ZZ intersections on the leftmost and middle figures.
Definition 4.5.

Let γ\gamma be an arc in some triangulation TT of (𝐒,𝐌)(\mathbf{S},\mathbf{M}), let LL be a lamination and let QγQ_{\gamma} is the local quadrilateral that γ\gamma is a diagonal of. The Shear coordinate of LL and γ\gamma with respect to TT, denoted bT(L,γ)b_{T}(L,\gamma), is given by

bT(L,γ)=#{S -intersections of L with Qγ}#{Z -intersections of L with Qγ}b_{T}(L,\gamma)=\#\{S\text{ -intersections of }L\text{ with }Q_{\gamma}\}-\#\{Z\text{ -intersections of }L\text{ with }Q_{\gamma}\}

where SS-(respectively ZZ-) intersections are illustrated in Figure 4.

We now use the notion of Shear coordinate to connect laminations to snake/band graphs and their perfect matchings. Namely, each diagonal or label of a tile of a snake/band graph corresponds to some arc in a triangulation. We use the following definition to assign a ±1\pm 1 to an orientation of a diagonal associated to a tile in a snake/band graph.

Definition 4.6.

Let 𝒢=(G1,,Gd)\mathcal{G}=(G_{1},\dots,G_{d}) be a snake (respectively band) graph and let PP be a perfect matching (respectively good perfect matching) of 𝒢\mathcal{G}. For each tile GiG_{i} of 𝒢\mathcal{G} labeled on its diagonal, PP induces an orientation on the diagonal of GiG_{i}. The orientation of ii is governed by the unique path from the SW vertex of G1G_{1} to the NE vertex of GdG_{d} taking alternating edges along PP and the diagonals of 𝒢\mathcal{G}.

Example 4.7.

Consider the following perfect matching of the snake graph highlighted in red in Figure 5. The orientation of each diagonal is designated by the path highlighted in blue. Note that this is also known in the literature as a TT-path defined in [10].

Figure 5. Orientation of diagonals induced by a perfect matching of a snake graph.

The orientation of each tile in a snake graph allows for the following definition:

Definition 4.8.

Let 𝒢=𝒢α,T\mathcal{G}=\mathcal{G}_{\alpha,T} be the snake (respectively band) graph associated to the curve α\alpha and triangulation TT of (𝐒,𝐌)(\mathbf{S},\mathbf{M}). Let T\mathcal{L}_{T} be a principal lamination of (𝐒,𝐌)(\mathbf{S},\mathbf{M}). A diagonal γik\gamma_{i_{k}} of 𝒢\mathcal{G} is T\mathcal{L}_{T}-oriented with respect to a perfect matching (respectively good perfect matching) PP if:

  • The tile is indexed odd and either:

    • bT(Lγik,γik)=1b_{T}(L_{\gamma_{i_{k}}},\gamma_{i_{k}})=1 and the diagonal on the tile is oriented down, or

    • bT(Lγik,γik)=1b_{T}(L_{\gamma_{i_{k}}},\gamma_{i_{k}})=-1 and the diagonal on the tile is oriented up.

  • The tile is indexed even and either:

    • bT(Lγik,γik)=1b_{T}(L_{\gamma_{i_{k}}},\gamma_{i_{k}})=1 and the diagonal on the tile is oriented up, or

    • bT(Lγik,γik)=1b_{T}(L_{\gamma_{i_{k}}},\gamma_{i_{k}})=-1 and the diagonal on the tile is oriented down.

This definition of T\mathcal{L}_{T}-oriented tells us exactly how to assign coefficients in our expansions.

Definition 4.9.

Given a (good) perfect matching PP of 𝒢\mathcal{G}, the coefficient monomial is given by

yT(P)=γik is T oriented yγiky_{\mathcal{L}_{T}}(P)=\prod_{\gamma_{i_{k}}\text{ is }\mathcal{L}_{T}\text{ oriented }}y_{\gamma_{i_{k}}}

With these coefficients, we state Wilson’s complete expansion formula, Theorem 5.44 in [11], following the setup from Theorem 3.7 for quasi-arcs:

Theorem 4.10.

[11] Let γ\gamma be a one-sided closed curve overlayed on a quasi-triangulation TT, let T\mathcal{L}_{T} be a principal lamination and 𝒫\mathcal{P} be the set of good perfect matchings of band graph 𝒢γ¯\mathcal{G}_{\overline{\gamma}}. The Laurent expansion for the quasi-cluster variable xγx_{\gamma} can be expressed as follows:

xγ=1bad(T,γ)1crossT(γ)P𝒫x(P)yT(P)x_{\gamma}=\frac{1}{\text{bad}(\mathcal{L}_{T},\gamma)}\frac{1}{\text{cross}_{T}(\gamma)}\sum_{P\in\mathcal{P}}x(P)y_{\mathcal{L}_{T}}(P)

where x(P)x(P) is the product of all the edge labels of 𝒢γ~\mathcal{G}_{\tilde{\gamma}} in PP, yT(P)y_{\mathcal{L}_{T}}(P) is the coefficient monomial and bad(T,γ)(\mathcal{L}_{T},\gamma) is an error term counting the number of “bad encounters” as in Definition 5.40 in [11].

5. Matrix Formulae on Orientable Surfaces with Respect to Principal Laminations

In this section, we present our main results in Theorem 5.8, Theorem 5.15, and then our main results in Theorem 6.4 and Theorem 6.8.

We generalize the matrix formulae from [8] so they work when using coefficients from an arbitrary principal lamination, rather than just principal coefficients, i.e. the case where bT(T,γ)=1b_{T}(\mathcal{L}_{T},\gamma)=1 for all γT\gamma\in T. This will be necessary when we eventually generalize these matrix formulae to non-orientable surfaces, for if we lift an arc α\alpha on a non-orientable surface to an orientable surface, it will lift to two arcs α1\alpha_{1} and α2\alpha_{2}, one of which will have an SS-intersection, while the other will have a ZZ-intersection. With this in mind, many of the results on cluster algebras found in this section will naturally carry over to quasi-cluster algebras due to the existence of the orientable double cover.

Fix a marked surface (𝐒,𝐌)(\mathbf{S},\mathbf{M}), triangulation TT of (𝐒,𝐌)(\mathbf{S},\mathbf{M}) and principal lamination T={Lγ|γT}\mathcal{L}_{T}=\{L_{\gamma}\,|\,\gamma\in T\}. To a generalized arc or closed loop γ\gamma, we will be associating a cluster algebra element xγx_{\gamma} using products of matrices in PSL2()PSL_{2}(\mathbb{R}). This can be achieved by recreating γ\gamma via a concatenation of various elementary steps, each of which has an associated matrix. The ensuing path created by adjoining these elementary steps will be referred to as an MM-path. Upon taking the product of these matrices, xγx_{\gamma} is either the upper right entry or the trace of the resulting matrix, depending on whether γ\gamma was a generalized curve or a closed loop, respectively. The expansion for xγx_{\gamma} achieved in this manner coincides with what one finds using more traditional means, such as mutation or the snake graph poset structure.

The definitions and results that we will be stating, and in some instances adjusting, from this section can be found in Sections 4 and 5 of [8]. In order to define the aforementioned elementary steps and their matrices, we first need to establish some preliminary notation.

Elementary steps do not go from marked point to marked point, rather, they go between points which are close to marked points, but are not marked points themselves. With the above in mind, for each marked point m𝐌m\in\mathbf{M}, we draw a small horocycle hmh_{m} locally around mm, and if mm is on the boundary, we just consider hmSh_{m}\cap S. We may assume the circles are small enough so that hmhm=h_{m}\cap h_{m^{\prime}}=\emptyset for distinct marked points m,mm,m^{\prime}. For each arc τT\tau\in T, and marked point m𝐌m\in\mathbf{M} incident to τ\tau, we let vm,τv_{m,\tau} denote the intersection point hmτh_{m}\cap\tau and we let vm,τ+v^{+}_{m,\tau} (resp. vm,τv^{-}_{m,\tau}) denote a point on hmh_{m} which is very close to vm,τv_{m,\tau} but in the clockwise (resp. counterclockwise) direction from vmv_{m}.

We now give the definition of elementary steps from [8] with a slight modification.

Definition 5.1 (Elementary Steps).
  • For the first type of step, we consider two arcs τ\tau and τ\tau^{\prime} from TT which are both incident to a marked point mm and which form a triangle with third side σT\sigma\in T. Then the first step is a curve that goes between τ\tau and τ\tau^{\prime} along hmh_{m}. The matrix associated to this step is [10±xσxτxτ1]\begin{bmatrix}1&0\\ \pm\frac{x_{\sigma}}{x_{\tau}x_{\tau^{\prime}}}&1\end{bmatrix} where the sign of xσxτxτ\frac{x_{\sigma}}{x_{\tau}x_{\tau^{\prime}}} is positive if the orientation is clockwise and it is negative otherwise.

  • For the second type of step, we cross τ\tau by following hmh_{m} between vm,τ+v^{+}_{m,\tau} and vm,τv^{-}_{m,\tau}. The associated matrix is [yτδ1,bT(T,τ)00yτδ1,bT(T,τ)]\begin{bmatrix}y_{\tau}^{\delta_{-1,b_{T}(\mathcal{L}_{T},\tau)}}&0\\ 0&y_{\tau}^{\delta_{1,b_{T}(\mathcal{L}_{T},\tau)}}\end{bmatrix} if we travel clockwise and [yτδ1,bT(T,τ)00yτδ1,bT(T,τ)]\begin{bmatrix}y_{\tau}^{\delta_{1,b_{T}(\mathcal{L}_{T},\tau)}}&0\\ 0&y_{\tau}^{\delta_{-1,b_{T}(\mathcal{L}_{T},\tau)}}\end{bmatrix} otherwise. Here, δ\delta is the Kronecker delta.

  • For the third type of step, we travel along a path parallel to a fixed arc τ\tau connecting two points vm,τ±v^{\pm}_{m,\tau} and vm,τv^{\mp}_{m^{\prime},\tau} associated to distinct marked points m,mm,m^{\prime}. The associated matrix is [0±xτ1xτ0]\begin{bmatrix}0&\pm x_{\tau}\\ \mp\frac{1}{x_{\tau}}&0\end{bmatrix}, where we use xr,1xrx_{r},\frac{-1}{x_{r}} if this step sees τ\tau on the right and xr,1xr-x_{r},\frac{1}{x_{r}} otherwise.

Remark 5.2.

The first and third elementary steps are identical to those in [8]. However, the second step is modified in order to reflect the underlying coefficient system using principal laminations.

With the adjusted type two step, we must formally restate and reprove the major results from Section 5 of [8] with the additional bT(T,τ)=1b_{T}(\mathcal{L}_{T},\tau)=-1 case in mind, as there are a few differences that manifest.

By concatenating these elementary step segments to make a path, we can associate a matrix to an arc or a loop in the following way.

Definition 5.3.

[8] Given (𝐒,𝐌),T(\mathbf{S},\mathbf{M}),T, and a generalized arc γ\gamma in (𝐒,𝐌)(\mathbf{S},\mathbf{M}) from ss to tt, we choose a curve ργ\rho_{\gamma} satisfying the following:

  • It begins at points of the form vs,τ±v^{\pm}_{s,\tau} and ends at points of the form vt,τ±v^{\pm}_{t,\tau^{\prime}} where τ,τ\tau,\tau^{\prime} are arcs of TT incident to ss and tt, respectively.

  • It is a concatenation of the elementary steps from Definition 5.1, and is isotopic to the segment of γ\gamma between hsγh_{s}\cap\gamma and htγh_{t}\cap\gamma.

  • The intersections of ργ\rho_{\gamma} with TT are in bijection with the intersections of γ\gamma with TT.

An analogous definition holds when γ\gamma is a closed loop as well, with the exception that we must have that ργ\rho_{\gamma} is isotopic to γ\gamma. In either instance, we refer to ργ\rho_{\gamma} as the MM-path. If we have a decomposition ργ=ρtρ1\rho_{\gamma}=\rho_{t}\circ\cdots\rho_{1} into elementary steps ρi\rho_{i}, then we define M(ργ)=M(ρt)M(ρ2)M(ρ1)M(\rho_{\gamma})=M(\rho_{t})\cdots M(\rho_{2})M(\rho_{1}), and the identity matrix is the matrix associated to the empty path.

Notation 5.4.

Given a 2×22\times 2 matrix M=(mij)M=(m_{ij}), let ur(M)\text{ur}(M) denote m12m_{12} (the upper-right entry of MM) and tr(M)\text{tr}(M) denote the trace of MM.

Remark 5.5 ([8]).

The matrices corresponding to elementary steps of type two are not in SL2()\text{SL}_{2}(\mathbb{R}). For some applications, we can instead use the matrices

[yτ1200yτ12]and[yτ1200yτ12].\begin{bmatrix}y_{\tau}^{-\frac{1}{2}}&&0\\ 0&&y_{\tau}^{\frac{1}{2}}\end{bmatrix}\qquad\text{and}\qquad\begin{bmatrix}y_{\tau}^{\frac{1}{2}}&&0\\ 0&&y_{\tau}^{-\frac{1}{2}}\end{bmatrix}.

If we make this replacement, we write M¯(ργ)=M¯(ρt)M¯(ρ2)M¯(ρ1)\overline{M}(\rho_{\gamma})=\overline{M}(\rho_{t})\cdots\overline{M}(\rho_{2})\overline{M}(\rho_{1}), which is a product of matrices in SL2()\text{SL}_{2}(\mathbb{R}).

While M(ργ)M(\rho_{\gamma}) and M¯(ργ)\overline{M}(\rho_{\gamma}) depend on their choice of ργ\rho_{\gamma}, it turns out that their trace and upper right entry depend only on γ\gamma.

Lemma 5.6.

[8, Lemma 4.8] Let γ1\gamma_{1} and γ2\gamma_{2} be a generalized arc and a closed loop, respectively, without contractible kinks. Then if ρ\rho and ρ\rho^{\prime} are two MM-paths associated to γ1\gamma_{1}, then

|ur(M((ρ))|=|ur(M(ρ))|.|\text{ur}(M((\rho))|=|\text{ur}(M(\rho^{\prime}))|.

Analogously, for any two MM-paths ρ,ρ\rho,\rho^{\prime} associated to γ2\gamma_{2}, we have

|tr(M(ρ))|=|tr(M(ρ))|.|\text{tr}(M(\rho))|=|\text{tr}(M(\rho^{\prime}))|.

The lemma above allows us to make the following definition.

Definition 5.7.

Let γ\gamma be a generalized arc and γ\gamma^{\prime} be closed loop, and let ρ\rho and ρ\rho^{\prime} denote arbitrary MM-paths associated to γ\gamma and γ\gamma^{\prime}, respectively. We associate the following algebraic quantities to γ\gamma and γ\gamma^{\prime}:

  1. (1)

    χγ,T=|ur(M(ρ))|\chi_{\gamma,\mathcal{L}_{T}}=|\text{ur}(M(\rho))| and χγ,T=|tr(M(ρ))|\chi_{\gamma^{\prime},\mathcal{L}_{T}}=|\text{tr}(M(\rho^{\prime}))|.

  2. (2)

    χ¯γ,T=|ur(M¯(ρ))|\overline{\chi}_{\gamma,\mathcal{L}_{T}}=|\text{ur}(\overline{M}(\rho))| and χ¯γ,T=|tr(M¯(ρ))|\overline{\chi}_{\gamma^{\prime},\mathcal{L}_{T}}=|\text{tr}(\overline{M}(\rho^{\prime}))|.

With the necessary definitions out of the way, we will now work towards proving the following result, which generalizes Theorem 4.11 of [8] so that our matrix formulae now account for any choice of principal lamination T\mathcal{L}_{T}.

Theorem 5.8.

Let (𝐒,𝐌)(\mathbf{S},\mathbf{M}) be marked surface with a triangulation TT and let T={τ1,,τn}T=\{\tau_{1},\ldots,\tau_{n}\} be the corresponding triangulation. Let 𝒜T(𝐒,𝐌)\mathcal{A}_{\mathcal{L}_{T}}(\mathbf{S},\mathbf{M}) be the corresponding cluster algebra associated to the principal lamination T\mathcal{L}_{T}.

  • Suppose γ\gamma is a generalized arc in SS without contractible kinks. Let 𝒢T,γ\mathcal{G}_{T,\gamma} be the snake graph of γ\gamma with respect to TT. Then

    χγ,T=1crossT(γ)Px(P)yT(P)\chi_{\gamma,\mathcal{L}_{T}}=\frac{1}{\text{cross}_{T}(\gamma)}\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)

    where the sum is over all perfect matchings PP of 𝒢T,γ\mathcal{G}_{T,\gamma}. It follows that when γ\gamma is an arc, then χγ,T\chi_{\gamma,\mathcal{L}_{T}} is the Laurent expansion of xγx_{\gamma} with respect to 𝒜T(𝐒,𝐌)\mathcal{A}_{\mathcal{L}_{T}}(\mathbf{S},\mathbf{M}).

  • Suppose that γ\gamma is a closed loop which is not contractible and has no contractible kinks. Then

    χγ,T=1crossTγ)Px(P)yT(P)\chi_{\gamma,\mathcal{L}_{T}}=\frac{1}{\text{cross}_{T}\gamma)}\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)

    where the sum is over all good matchings PP of the band graph 𝒢~T,γ\widetilde{\mathcal{G}}_{T,\gamma}. Again, χγ,T\chi_{\gamma,\mathcal{L}_{T}} is the Laurent expansion of γ\gamma with respect to 𝒜T(𝐒,𝐌)\mathcal{A}_{\mathcal{L}_{T}}(\mathbf{S},\mathbf{M}).

To prove Theorem 5.8, we will show χγ,T\chi_{\gamma,\mathcal{L}_{T}} coincides with the perfect (good) matching enumerator associated to their respective snake (band) graph. The process will be similar to that of [8]; however, with the additional bT(γ,T)=1b_{T}(\gamma,\mathcal{L}_{T})=-1 case, we will need to make a few changes to the set-up as well as a few adjustments to the statements of the results. The proof of Theorem 5.8 can be found at the end of Section 5.2.

5.1. Matchings of Snake and Band Graphs

To begin the proof, we will associate 2×22\times 2 matrices to the parallelograms of a snake graph, which will then give us a way of representing the Laurent expansion of our graph in terms of products of matrices. The exposition defining the snake and band graphs is taken from [8].

Definition 5.9 (Abstract snake graph).

An abstract snake graph with dd tiles is formed by concatenating the following pieces:

  • An initial triangle

    i1\displaystyle i_{1}a\displaystyle ab\displaystyle b
  • d1d-1 parallelograms H1,,Hd1H_{1},\ldots,H_{d-1}, where each HjH_{j} is either a north facing or east-pointing parallelogram:

    ij+1\displaystyle i_{j+1}ij\displaystyle i_{j}aj\displaystyle a_{j}ij\displaystyle i_{j}ij+1\displaystyle i_{j+1}ij\displaystyle i_{j}ij\displaystyle i_{j}ij+1\displaystyle i_{j+1}ij+1\displaystyle i_{j+1}aj\displaystyle a_{j}North-pointingEast-pointing
  • A final triangle based on whether dd is odd or even

    id\displaystyle i_{d}z\displaystyle zw\displaystyle wid\displaystyle i_{d}w\displaystyle wz\displaystyle zEvenOdd

We then erase all diagonal edges from the figure.

Definition 5.10 (Abstract band graph).

An abstract band graph for a one-sided curve with dd tiles is formed by concatenating the following puzzle pieces:

  • An initial triangle

    a\displaystyle ax\displaystyle xy\displaystyle yi1\displaystyle i_{1}id\displaystyle i_{d}
  • d1d-1 parallelograms H1,,Hd1H_{1},\ldots,H_{d-1}, where each HjH_{j} is as before.

  • A final triangle based on whether dd is odd or even.

    id\displaystyle i_{d}i1\displaystyle i_{1}a\displaystyle a^{\prime}id\displaystyle i_{d}a\displaystyle a^{\prime}i1\displaystyle i_{1}evenoddx\displaystyle x^{\prime}y\displaystyle y^{\prime}x\displaystyle x^{\prime}y\displaystyle y^{\prime}

We then identify the edges aa and aa^{\prime}, the vertices xx and xx^{\prime}, and the vertices yy and yy^{\prime}. Lastly, we erase all diagonal edges from the figure.

Just like a traditional snake and band graph, we can consider perfect matchings of an abstract snake graph and good matchings of an abstract band graph. Furthermore, we can associate to each perfect/good matching its weight and coefficient monomials x(P)x(P) and yT(P)y_{\mathcal{L}_{T}(P)} by letting T={τi1,,τid}T=\{\tau_{i_{1}},\ldots,\tau_{i_{d}}\}.

Definition 5.11.

Let GG be an abstract snake or band graph with dd tiles. For each parallelogram HiH_{i} of GG, we associate a 2×22\times 2 matrix mim_{i}, where mim_{i} is either

[10xajxijxij+1yij]or[xij+1xijxajyij0xijyjxij+1] when bT(T,ij)=1\begin{bmatrix}1&&0\\ \frac{x_{a_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&y_{i_{j}}\end{bmatrix}\quad\text{or}\quad\begin{bmatrix}\frac{x_{i_{j+1}}}{x_{i_{j}}}&&x_{a_{j}}y_{i_{j}}\\ 0&&\frac{x_{i_{j}}y_{j}}{x_{i_{j+1}}}\end{bmatrix}\quad\text{ when }b_{T}(\mathcal{L}_{T},i_{j})=1
[yij0xajyijxijxij+11]or[xij+1yijxijxaj0xijxij+1] when bT(T,ij)=1\begin{bmatrix}y_{i_{j}}&&0\\ \frac{x_{a_{j}}y_{i_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&1\end{bmatrix}\quad\text{or}\quad\begin{bmatrix}\frac{x_{i_{j+1}}y_{i_{j}}}{x_{i_{j}}}&&x_{a_{j}}\\ 0&&\frac{x_{i_{j}}}{x_{i_{j+1}}}\end{bmatrix}\quad\text{ when }b_{T}(\mathcal{L}_{T},i_{j})=-1

In either case, the following conditions determine which of the two matrices we use:

  • m1m_{1} is of the first type if H1H_{1} is a north-pointing parallelogram, and otherwise it is of the second type.

  • for i>1i>1, mim_{i} is of the first type if both Hi1H_{i-1} and HiH_{i} have the same shape, and otherwise, it is of the second type.

We can then define a matrix MdM_{d} associated to GG defined by Md=md1m1M_{d}=m_{d-1}\cdot\cdots\cdot m_{1} when d>1d>1. Otherwise, M1=I2M_{1}=I_{2}.

Remark 5.12.

Definition 5.11 differs from [8] as it depends on the particular choice of Shear coordinate. This is necessary to reflect the coefficient system provided by the principal laminations in the non-orientable setting.

The upcoming sequence of results generalize Proposition 5.5, Corollary 5.6 and Theorem 5.4 from [8] to principal laminations. Despite the additional case that stems from having ZZ-shape intersections, the proof techniques largely stay the same and the statements generalize in an expected manner.

Proposition 5.13.

Let GG be an abstract snake graph with dd tiles. Write

Md=[AdBdCdDd].M_{d}=\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}.

For d2d\geq 2,

  • When bT(T,id1)=1b_{T}(\mathcal{L}_{T},{i_{d-1}})=1 we have

    Ad=PSAx(P)yT(P)(xi1xid1)xaxw\displaystyle A_{d}=\frac{\sum_{P\in S_{A}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{1}}\cdots x_{i_{d-1}})x_{a}x_{w}} Bd=PSBx(P)yT(P)(xi2xid1)xbxw\displaystyle\qquad B_{d}=\frac{\sum_{P\in S_{B}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d-1}})x_{b}x_{w}}
    Cd=PSCx(P)yT(P)(xixid)xaxzyid\displaystyle C_{d}=\frac{\sum_{P\in S_{C}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i}\cdots x_{i_{d}})x_{a}x_{z}y_{i_{d}}} Dd=PSDx(P)yT(P)(xi2xid)xbxzyid\displaystyle\qquad D_{d}=\frac{\sum_{P\in S_{D}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d}})x_{b}x_{z}y_{i_{d}}}
  • When bT(T,id1)=1b_{T}(\mathcal{L}_{T},i_{d-1})=-1 we have

    Ad=PSAx(P)yT(P)(xi1xid1)xaxwyid\displaystyle A_{d}=\frac{\sum_{P\in S_{A}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{1}}\cdots x_{i_{d-1}})x_{a}x_{w}y_{i_{d}}} Bd=PSBx(P)yT(P)(xi2xid1)xbxwyid\displaystyle\qquad B_{d}=\frac{\sum_{P\in S_{B}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d-1}})x_{b}x_{w}y_{i_{d}}}
    Cd=PSCx(P)yT(P)(xixid)xaxz\displaystyle C_{d}=\frac{\sum_{P\in S_{C}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i}\cdots x_{i_{d}})x_{a}x_{z}} Dd=PSDx(P)yT(P)(xi2xid)xbxz\displaystyle\qquad D_{d}=\frac{\sum_{P\in S_{D}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d}})x_{b}x_{z}}

For d=1d=1, the formulae remain the same, but the cases are determined by the sign of bT(T,i1)b_{T}(\mathcal{L}_{T},i_{1}).

Here, SA,SB,SCS_{A},S_{B},S_{C} and SDS_{D} are the sets of perfect matchings of GG which use the edges {a,w},{b,w},{a,z}\{a,w\},\{b,w\},\{a,z\} and {b,z}\{b,z\}, respectively.

Proof.

The case where bT(T,id1)=1b_{T}(\mathcal{L}_{T},{i_{d-1}})=1 is exactly Proposition 5.5 in [8], so we only consider the case where bT(T,id1)=1b_{T}(\mathcal{L}_{T},{i_{d-1}})=-1. Additionally, we note that differences between the two cases will only involve changes in yT(P)y_{\mathcal{L}_{T}}(P). This is due to the fact that, despite the change in principal lamination, the perfect matchings on the abstract snake graph remain the exact same, meaning x(P)x(P) will not change when flipping from a SS-shape intersection to a ZZ-shape intersection, or vice versa.

We perform a proof by induction on dd, and consider what happens when one adds one more tile to a snake graph. When d=1d=1, we just have a single tile

aabbwwzzi1i_{1}

meaning SA={a,w},SB=SC=S_{A}=\{a,w\},S_{B}=S_{C}=\emptyset and SD={b,z}S_{D}=\{b,z\}. When bT(T,i1)=1b_{T}(\mathcal{L}_{T},i_{1})=-1, we get A1=xaxwyi1xaxwyi1,B1=0,C1=0,D1=xbxzxbxzA_{1}=\frac{x_{a}x_{w}y_{i_{1}}}{x_{a}x_{w}y_{i_{1}}},\\ B_{1}=0,C_{1}=0,D_{1}=\frac{x_{b}x_{z}}{x_{b}x_{z}}, hence m1=I2m_{1}=I_{2}, as desired.

Just like Proposition 5.5 in [8], for d>1d>1, we let GG^{\prime} denote the graph obtained by gluing together the initial triangle, parallelograms H1,,Hd2H_{1},\ldots,H_{d-2} and the final triangle. For convenience, we label the final triangle in GG^{\prime} with ww^{\prime} and zz^{\prime} and note the orientation of this triangle depends on whether (d1)(d-1) is odd or even. By changing labels, the graph GG^{\prime} is isomorphic to the subgraph of GG consisting of the first d1d-1 tiles. In particular, we either replace the edge label ww^{\prime} with ad1a_{d-1} and zz with idi_{d}, or vice versa.

With this in mind, when considering d=2d=2, we have may assume it has the following shape and labeling:

aabbww^{\prime}i2=zi_{2}=z^{\prime}i1i_{1}i1i_{1}zzwwi2i_{2}

One can quickly check that when bT(T,i1)=1b_{T}(\mathcal{L}_{T},i_{1})=1 and bT(T,i2)=±1b_{T}(\mathcal{L}_{T},i_{2})=\pm 1 then M2=[10xwxi1xi2yi1]M_{2}=\begin{bmatrix}1&&0\\ \frac{x_{w^{\prime}}}{x_{i_{1}}x_{i_{2}}}&&y_{i_{1}}\end{bmatrix} meanwhile, when bT(T,i1)=1b_{T}(\mathcal{L}_{T},i_{1})=-1 and bT(T,i2)=±1b_{T}(\mathcal{L}_{T},i_{2})=\pm 1 then M2=[yi10xwyi1xi1xi21]M_{2}=\begin{bmatrix}y_{i_{1}}&&0\\ \frac{x_{w^{\prime}}y_{i_{1}}}{x_{i_{1}}x_{i_{2}}}&&1\end{bmatrix}. Regardless, both matrices match the equations found in Definition 5.11.

To finish the proof, we simply observe that the bijections between the various perfect matchings utilized in the proof of Proposition 5.5 of [8] apply here as well, regardless of our principal lamination. As such, by applying induction, we may conclude the following. Under the initial labeling scheme (used in the base cases), we obtain the following two cases depending on the sign of bT(T,id1)b_{T}(\mathcal{L}_{T},i_{d-1}):

[AdBdCdDd]\displaystyle\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix} =[10xad1xid1xidyid1][Ad1Bd1Cd1Dd1]\displaystyle=\begin{bmatrix}1&&0\\ \frac{x_{a_{d-1}}}{x_{i_{d-1}}x_{i_{d}}}&&y_{i_{d-1}}\end{bmatrix}\begin{bmatrix}A_{d-1}&&B_{d-1}\\ C_{d-1}&&D_{d-1}\end{bmatrix}
[AdBdCdDd]\displaystyle\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix} =[yid10xad1yid1xid1xid1][Ad1Bd1Cd1Dd1]\displaystyle=\begin{bmatrix}y_{i_{d-1}}&&0\\ \frac{x_{a_{d-1}}y_{i_{d-1}}}{x_{i_{d-1}}x_{i_{d}}}&&1\end{bmatrix}\begin{bmatrix}A_{d-1}&&B_{d-1}\\ C_{d-1}&&D_{d-1}\end{bmatrix}

Likewise, under the other labeling scheme we obtain the following cases, again, depending on the sign of bT(T,id1)b_{T}(\mathcal{L}_{T},i_{d-1}):

[AdBdCdDd]\displaystyle\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix} =[xidxid1xad1yid10xid1yid1xid][Ad1Bd1Cd1Dd1]\displaystyle=\begin{bmatrix}\frac{x_{i_{d}}}{x_{i_{d-1}}}&&x_{a_{d-1}}y_{i_{d-1}}\\ 0&&\frac{x_{i_{d-1}}y_{i_{d-1}}}{x_{i_{d}}}\end{bmatrix}\begin{bmatrix}A_{d-1}&&B_{d-1}\\ C_{d-1}&&D_{d-1}\end{bmatrix}
[AdBdCdDd]\displaystyle\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix} =[xidyd1xid1xad10xid1xid][Ad1Bd1Cd1Dd1]\displaystyle=\begin{bmatrix}\frac{x_{i_{d}}y_{d-1}}{x_{i_{d-1}}}&&x_{a_{d-1}}\\ 0&&\frac{x_{i_{d-1}}}{x_{i_{d}}}\end{bmatrix}\begin{bmatrix}A_{d-1}&&B_{d-1}\\ C_{d-1}&&D_{d-1}\end{bmatrix}

Comparing these two equations with Definition 5.11, we see the various cases agree with the cases of the definition of MdM_{d}. ∎

Corollary 5.14.

Let GG be an abstract snake graph with dd tiles. Write Md=[AdBdCdDd]M_{d}=\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}.

  • When bT(T,id)=1b_{T}(\mathcal{L}_{T},i_{d})=1, then

    Px(P)yT(P)xi1xid\displaystyle\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{x_{i_{1}}\cdots x_{i_{d}}} =xaxwAdxid+xbxwBdxi1xid+xaxzyidCd+xbxzyidDdxi1\displaystyle=\frac{x_{a}x_{w}A_{d}}{x_{i_{d}}}+\frac{x_{b}x_{w}B_{d}}{x_{i_{1}}x_{i_{d}}}+x_{a}x_{z}y_{i_{d}}C_{d}+\frac{x_{b}x_{z}y_{i_{d}}D_{d}}{x_{i_{1}}}
    =ur([xwxidxzyid1xz0][AdBdCdDd][0xa1xaxbxi1])\displaystyle=\text{ur}(\begin{bmatrix}\frac{x_{w}}{x_{i_{d}}}&&x_{z}y_{i_{d}}\\ -\frac{1}{x_{z}}&&0\end{bmatrix}\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&\frac{x_{b}}{x_{i_{1}}}\end{bmatrix}) (5.1)
  • When bT(T,id)=1b_{T}(\mathcal{L}_{T},i_{d})=-1, then

    Px(P)yT(P)xi1xid\displaystyle\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{x_{i_{1}}\cdots x_{i_{d}}} =xaxwyidAdxid+xbxwyidBdxi1xid+xaxzCd+xbxzDdxi1\displaystyle=\frac{x_{a}x_{w}y_{i_{d}}A_{d}}{x_{i_{d}}}+\frac{x_{b}x_{w}y_{i_{d}}B_{d}}{x_{i_{1}}x_{i_{d}}}+x_{a}x_{z}C_{d}+\frac{x_{b}x_{z}D_{d}}{x_{i_{1}}}
    =ur([xwyidxidxzyidxz0][AdBdCdDd][0xa1xaxbxi1])\displaystyle=\text{ur}(\begin{bmatrix}\frac{x_{w}y_{i_{d}}}{x_{i_{d}}}&&x_{z}\\ -\frac{y_{i_{d}}}{x_{z}}&&0\end{bmatrix}\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&\frac{x_{b}}{x_{i_{1}}}\end{bmatrix}) (5.2)

where the sum is over all perfect matchings of GG.

The above corollary immediately implies the next theorem.

Theorem 5.15.

Suppose GG is an abstract snake graph with dd tiles. Then its perfect matching enumerator is given by

  • Px(P)yT(P)=xi1xidur([xwxidxzyid1xz0]Md[0xa1xaxbxi1]), when bT(T,id)=1\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)=x_{i_{1}}\cdots x_{i_{d}}\text{ur}(\begin{bmatrix}\frac{x_{w}}{x_{i_{d}}}&&x_{z}y_{i_{d}}\\ -\frac{1}{x_{z}}&&0\end{bmatrix}M_{d}\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&\frac{x_{b}}{x_{i_{1}}}\end{bmatrix}),\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1

  • Px(P)yT(P)=xi1xidur([xwyidxidxzyidxz0]Md[0xa1xaxbxi1]), when bT(T,id)=1\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)=x_{i_{1}}\cdots x_{i_{d}}\text{ur}(\begin{bmatrix}\frac{x_{w}y_{i_{d}}}{x_{i_{d}}}&&x_{z}\\ -\frac{y_{i_{d}}}{x_{z}}&&0\end{bmatrix}M_{d}\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&\frac{x_{b}}{x_{i_{1}}}\end{bmatrix}),\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1

By adjusting our labeling by substituting i1i_{1} for ww, aa^{\prime} for zz, and idi_{d} for bb (see Definition 5.10), we obtain a sequence of statements and results for abstract band graphs that will be analogous to Proposition 5.7 and Corollary 5.8 of [8], but with respect to principal laminations. We end up with the following theorem.

Theorem 5.16.

Suppose GG is an abstract band graph with dd tiles. Then its good matching enumerator i.e. the weighted sum of good perfect matchings of GG with respect to T\mathcal{L}_{T} is given by

  • Px(P)yT(P)=xi1xidtr([xi1xidxayid0xidyidxi1]Md), when bT(T,id)=1\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)=x_{i_{1}}\cdots x_{i_{d}}\text{tr}(\begin{bmatrix}\frac{x_{i_{1}}}{x_{i_{d}}}&&x_{a}y_{i_{d}}\\ 0&&\frac{x_{i_{d}}y_{i_{d}}}{x_{i_{1}}}\end{bmatrix}M_{d}),\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1

  • Px(P)yT(P)=xi1xidtr([xi1yidxidxa0xidxi1]Md), when bT(T,id)=1\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)=x_{i_{1}}\cdots x_{i_{d}}\text{tr}(\begin{bmatrix}\frac{x_{i_{1}}y_{i_{d}}}{x_{i_{d}}}&&x_{a}\\ 0&&\frac{x_{i_{d}}}{x_{i_{1}}}\end{bmatrix}M_{d}),\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1

5.2. The Standard MM-path

Arcs do not have a unique associated MM-path, but there is an algorithm for assigning a “standard MM-path” ργ\rho_{\gamma} to a curve γ\gamma that we will utilize in order to facilitate proofs. Here we recall the definition of a standard MM-path for both an arc and a closed loop. Despite working with principal laminations for coefficients, the standard MM-paths for generalized arcs and closed curves will not change. The only difference that will occur on the MM-path side will involve adjusting our step two matrix, depending on whether the arc’s corresponding lamination is an SS-intersection or a ZZ-intersection.

τij\displaystyle\tau_{i_{j}}τij+1\displaystyle\tau_{i_{j+1}}τij\displaystyle\tau_{i_{j}}τij+1\displaystyle\tau_{i_{j+1}}aj\displaystyle a_{j}aj\displaystyle a_{j}orCounterclockwiseClockwise
Figure 6. Possible transitions between τij\tau_{i_{j}} and τij+1\tau_{i_{j+1}} in the standard MM-path.
Definition 5.17 (Standard MM-paths).

The definitions below are Definitions 5.9 and 5.11 from [8]. In Definition 6.7, we generalize this construction for one-sided closed curves.

Generalized Arcs: We begin with the case where we have a generalized arc γ\gamma. Let γ\gamma be a generalized arc that goes from point PP to a point QQ, crossing the arcs τi1,,τid\tau_{i_{1}},\ldots,\tau_{i_{d}} in order. We label the initial triangle Δ0\Delta_{0} that intersects γ\gamma with sides a,ba,b, and τi1\tau_{i_{1}} in clockwise order so that PP is the intersection of aa and bb. We label the final crossed triangle Δd\Delta_{d} with sides w,zw,z and τid\tau_{i_{d}} in clockwise order, with QQ being the intersection of the arcs ww and zz.

The path ργ\rho_{\gamma} begins at the point vP,a±v^{\pm}_{P,a}, where the sign is chosen so the point lies in the triangle. The first two steps of ργ\rho_{\gamma} consists of traveling along aa and then jumping from aa to τi1\tau_{i_{1}}, i.e., we perform a step three and then a step one move. After these two moves, we are at a point of the form v,τi1±v^{\pm}_{*,\tau_{i_{1}}}, but we have not yet crossed τi1\tau_{i_{1}}.

Until we reach the the final triangle, we follow the steps illustrated in Figure 6 depending on whether τij+1\tau_{i_{j+1}} is counterclockwise or clockwise from τij\tau_{i_{j}}. That is, if τij+1\tau_{i_{j+1}} is counterclockwise to τij\tau_{i_{j}}, the next steps of ργ\rho_{\gamma} will consist of a step two move that crosses τij\tau_{i_{j}} and then a step one move from τij\tau_{i_{j}} to τij+1\tau_{i_{j+1}}. If τij+1\tau_{i_{j+1}} is clockwise from τij\tau_{i_{j}}, we cross τij\tau_{i_{j}} with a step two move, then we reach τij+1\tau_{i_{j+1}} through a step one move to reach aja_{j}, then a step three move to travel along aja_{j} and we finish with a step one move to arrive at τij\tau_{i_{j}}.

After (d1)(d-1) transitions, we reach the final triangle. We cross τid\tau_{i_{d}} with a step three move, we then apply a step one move to get to zz and then we travel along zz with a step three move to reach the point vQ,z±v^{\pm}_{Q,z}. We call this the standard MM-path associated to γ\gamma.

Closed Loops: In the case where γ\gamma is a closed loop, the process is largely the same. To begin, we choose a triangle Δ\Delta in TT such that two of its arcs are crossed by γ\gamma and we label the two arcs τi1\tau_{i_{1}} and τid\tau_{i_{d}} so that τi1\tau_{i_{1}} is clockwise from τid\tau_{i_{d}}. We label the third side of Δ\Delta by aa. Let pp be a point on γ\gamma which lies in Δ\Delta and has the form v,τi1±v^{\pm}_{*,\tau_{i_{1}}}. We then let τi1,,τid\tau_{i_{1}},\ldots,\tau_{i_{d}} denote the ordered sequence of arcs which are crossed by γ\gamma, when one travels from pp away from Δ\Delta. The standard MM-path ργ\rho_{\gamma} associated to γ\gamma is defined just like it was for generalized arcs, that is, by following the sequences of steps illustrated by Figure 6, depending on whether τij+1\tau_{i_{j+1}} is counterclockwise or clockwise from τij\tau_{i_{j}}. The standard MM-path begins and ends at the point pp and we consider the indices modulo nn.

If ργ\rho_{\gamma} is not a standard MM-path, for the generalized arc or loop γ\gamma, then one can deform γ\gamma into a standard MM-path ργ\rho_{\gamma} by making local adjustments that do not influence the upper right entry or trace.

We are now ready to prove Theorem 5.8, which states that, with our additional adjustments, the matrix formulae of [8] hold for arbitrary principal laminations.

Proof of Theorem 5.8.

We first consider the case where γ\gamma is a generalized arc and consider its standard MM-path, ργ\rho_{\gamma}. The first two steps of this path correspond to the matrix product

[10xbxaxi11][0xa1xa0]=[0xa1xaxbxi1]\begin{bmatrix}1&&0\\ \frac{x_{b}}{x_{a}x_{i_{1}}}&&1\end{bmatrix}\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&0\end{bmatrix}=\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&\frac{x_{b}}{x_{i_{1}}}\end{bmatrix}

while the last three steps correspond to

[0xz1xz0][10xwxidxz1][100yid]\displaystyle\begin{bmatrix}0&&x_{z}\\ -\frac{1}{x_{z}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{w}}{x_{i_{d}}x_{z}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ 0&&y_{i_{d}}\end{bmatrix} =[xwxidxzyid1xz0] when bT(T,id)=1\displaystyle=\begin{bmatrix}\frac{x_{w}}{x_{i_{d}}}&&x_{z}y_{i_{d}}\\ -\frac{1}{x_{z}}&&0\end{bmatrix}\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1
[0xz1xz0][10xwxidxz1][yid001]\displaystyle\begin{bmatrix}0&&x_{z}\\ -\frac{1}{x_{z}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{w}}{x_{i_{d}}x_{z}}&&1\end{bmatrix}\begin{bmatrix}y_{i_{d}}&&0\\ 0&&1\end{bmatrix} =[xwyidxidxzyidxz0] when bT(T,id)=1\displaystyle=\begin{bmatrix}\frac{x_{w}y_{i_{d}}}{x_{i_{d}}}&&x_{z}\\ -\frac{y_{i_{d}}}{x_{z}}&&0\end{bmatrix}\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1

In between the first and the last steps, for the portion between τij\tau_{i_{j}} and τij+1\tau_{i_{j+1}}, where 1jd11\leq j\leq d-1, we have the following four cases that depend on the sign of bT(T,τij)b_{T}(\mathcal{L}_{T},\tau_{i_{j}}) and whether τij+1\tau_{i_{j+1}} lies clockwise or counterclockwise from τij\tau_{i_{j}}.

[10xajxijxij+11][100yij]=[10xajxijxij+1yij](when bT(T,ij)=1,CCW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{a_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ 0&&y_{i_{j}}\end{bmatrix}=\begin{bmatrix}1&&0\\ \frac{x_{a_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&y_{i_{j}}\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=1,\text{CCW})
[10xajxijxij+11][yij001]=[yij0xajyijxijxij+11](when bT(T,ij)=1,CCW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{a_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&1\end{bmatrix}\begin{bmatrix}y_{i_{j}}&&0\\ 0&&1\end{bmatrix}=\begin{bmatrix}y_{i_{j}}&&0\\ \frac{x_{a_{j}}y_{i_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&1\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=-1,\text{CCW})
[10xijxajxij+11][0xaj1xaj0][10xij+1xajxij1][100yij]=[xij+1xijxajyij0xijyijxij+1](when bT(T,ij)=1,CW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{i_{j}}}{x_{a_{j}}x_{i_{j+1}}}&&1\\ \end{bmatrix}\begin{bmatrix}0&&x_{a_{j}}\\ -\frac{1}{x_{a_{j}}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{i_{j+1}}}{x_{a_{j}}x_{i_{j}}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ 0&&y_{i_{j}}\end{bmatrix}=\begin{bmatrix}\frac{x_{i_{j+1}}}{x_{i_{j}}}&&x_{a_{j}}y_{i_{j}}\\ 0&&\frac{x_{i_{j}}y_{i_{j}}}{x_{i_{j+1}}}\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=1,\text{CW})
[10xijxajxij+11][0xaj1xaj0][10xij+1xajxij1][yij001]=[xij+1yijxijxaj0xijxij+1](when bT(T,ij)=1,CW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{i_{j}}}{x_{a_{j}}x_{i_{j+1}}}&&1\\ \end{bmatrix}\begin{bmatrix}0&&x_{a_{j}}\\ -\frac{1}{x_{a_{j}}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{i_{j+1}}}{x_{a_{j}}x_{i_{j}}}&&1\end{bmatrix}\begin{bmatrix}y_{i_{j}}&&0\\ 0&&1\end{bmatrix}=\begin{bmatrix}\frac{x_{i_{j+1}}y_{i_{j}}}{x_{i_{j}}}&&x_{a_{j}}\\ 0&&\frac{x_{i_{j}}}{x_{i_{j+1}}}\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=-1,\text{CW})

Here, aja_{j} is the third side of the triangle with sides τij,τij+1\tau_{i_{j}},\tau_{i_{j+1}}.

By Definition 5.7 and Theorem 5.15, we find

  • χγ,T=ur([xwxidxzyid1xz0]M[0xa1xaxbxi1])=Px(P)yT(P)crossT(γ), when bT(T,id)=1\chi_{\gamma,\mathcal{L}_{T}}=\text{ur}(\begin{bmatrix}\frac{x_{w}}{x_{i_{d}}}&&x_{z}y_{i_{d}}\\ -\frac{1}{x_{z}}&&0\end{bmatrix}M\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&\frac{x_{b}}{x_{i_{1}}}\end{bmatrix})=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\gamma)},\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1

  • χγ,T=ur([xwyidxidxzyidxz0]M[0xa1xaxbxi1])=Px(P)yT(P)crossT(γ), when bT(T,id)=1\chi_{\gamma,\mathcal{L}_{T}}=\text{ur}(\begin{bmatrix}\frac{x_{w}y_{i_{d}}}{x_{i_{d}}}&&x_{z}\\ -\frac{y_{i_{d}}}{x_{z}}&&0\end{bmatrix}M\begin{bmatrix}0&&x_{a}\\ -\frac{1}{x_{a}}&&\frac{x_{b}}{x_{i_{1}}}\end{bmatrix})=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\gamma)},\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1

where the middle matrix MM is obtained by multiplying together a sequence of matrices of the four forms mentioned previously. The term χγ,T\chi_{\gamma,\mathcal{L}_{T}} has an interpretation in terms of perfect matchings of some abstract snake graph GG. Namely, the abstract snake graph GG is precisely 𝒢T,γ\mathcal{G}_{T,\gamma} associated to γ\gamma, meaning M=MdM=M_{d} from Definition 5.11, which justifies the right-most equality and the proof is complete.

The case where γ\gamma is a closed loop is virtually the same, the only difference being that our initial and final steps are different. By construction of the standard MM-path, we have that τi1\tau_{i_{1}} is in the clockwise direction of τid\tau_{i_{d}}. This implies the final steps of ργ\rho_{\gamma} corresponds to either

[xi1xidxayid0xidyidxi1] when bT(T,γ)=1\displaystyle\begin{bmatrix}\frac{x_{i_{1}}}{x_{i_{d}}}&&x_{a}y_{i_{d}}\\ 0&&\frac{x_{i_{d}}y_{i_{d}}}{x_{i_{1}}}\end{bmatrix}\text{ when }b_{T}(\mathcal{L}_{T},\gamma)=1
[xi1yidxidxa0xidxi1] when bT(T,γ)=1\displaystyle\begin{bmatrix}\frac{x_{i_{1}}y_{i_{d}}}{x_{i_{d}}}&&x_{a}\\ 0&&\frac{x_{i_{d}}}{x_{i_{1}}}\end{bmatrix}\text{ when }b_{T}(\mathcal{L}_{T},\gamma)=-1

By Definition 5.7, Theorem 5.16 and using the interpretation of χγ,T\chi_{\gamma,\mathcal{L}_{T}} in terms of good matchings of some abstract band graph GG, we obtain

  • χγ,T=tr([xi1xidxayid0xidyidxi1]M)=Px(P)yT(P)crossT(γ), when bT(T,id)=1\chi_{\gamma,\mathcal{L}_{T}}=\text{tr}(\begin{bmatrix}\frac{x_{i_{1}}}{x_{i_{d}}}&&x_{a}y_{i_{d}}\\ 0&&\frac{x_{i_{d}}y_{i_{d}}}{x_{i_{1}}}\end{bmatrix}M)=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\gamma)},\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1

  • χγ,T=tr([xi1yidxidxa0xidxi1]M)=Px(P)yT(P)crossT(γ), when bT(T,id)=1\chi_{\gamma,\mathcal{L}_{T}}=\text{tr}(\begin{bmatrix}\frac{x_{i_{1}}y_{i_{d}}}{x_{i_{d}}}&&x_{a}\\ 0&&\frac{x_{i_{d}}}{x_{i_{1}}}\end{bmatrix}M)=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\gamma)},\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1

where again MM is a obtained by multiplying together a sequence of matrices of the four forms mentioned previously and can be interpreted as the matrix MdM_{d} for 𝒢¯T,γ\overline{\mathcal{G}}_{T,\gamma}. ∎

5.3. Example with Closed Curve

Consider the triangulation and closed loop γ\gamma with band graph 𝒢~T,γ\widetilde{\mathcal{G}}_{T,\gamma} pictured in Figure 7.

γ\gammaτ1\tau_{1}τ2\tau_{2}τ4\tau_{4}τ3\tau_{3}b1b_{1}b3b_{3}b4b_{4}b2b_{2}b2b_{2}b1b_{1}i2i_{2}i4i_{4}i1i_{1}yyxxyy^{\prime}xx^{\prime}i4i_{4}i3i_{3}i2i_{2}i3i_{3}i1i_{1}i4i_{4}b2b_{2}i2i_{2}i3i_{3}
Figure 7. Triangulation TT and band graph 𝒢~T,γ\widetilde{\mathcal{G}}_{T,\gamma} for the closed loop γ\gamma

We compute χγ,T\chi_{\gamma,\mathcal{L}_{T}^{\prime}} as well as Px(P)yT(P)crossT(γ)\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}^{\prime}}(P)}{\text{cross}_{T}(\gamma)} and show they are equal. The perfect matchings and how they relate to one another are pictured in Figure 8. Taking the sum of each monomial in the Figure 8, we find

Px(P)yT(P)crossT(γ)=y2x12x2x4+y2y3x12+(y3+y2y3y4)x1x3+y3y4x32+y1y3y4x2x32x4x1x2x3x4\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}^{\prime}}(P)}{\text{cross}_{T}(\gamma)}=\frac{y_{2}x_{1}^{2}x_{2}x_{4}+y_{2}y_{3}x_{1}^{2}+(y_{3}+y_{2}y_{3}y_{4})x_{1}x_{3}+y_{3}y_{4}x_{3}^{2}+y_{1}y_{3}y_{4}x_{2}x_{3}^{2}x_{4}}{x_{1}x_{2}x_{3}x_{4}}
Refer to caption
Figure 8. Perfect matchings on 𝒢~T,γ\widetilde{\mathcal{G}}_{T,\gamma}, along with their associated weight monomial. Note, yT(P)y_{\mathcal{L}_{T}^{\prime}}(P) is calculated using Definition 4.8 for each perfect matching PP

Meanwhile, the matrix formula yields

χT,γ\displaystyle\chi_{\mathcal{L}_{T}^{\prime},\gamma} =tr([x1x4y4b20x4y4x1][x4x3y3b30x3y3x4][y20b4y2x2x31][10b1x1x2y1])\displaystyle=\text{tr}(\begin{bmatrix}\frac{x_{1}}{x_{4}}&&y_{4}b_{2}\\ 0&&\frac{x_{4}y_{4}}{x_{1}}\end{bmatrix}\begin{bmatrix}\frac{x_{4}}{x_{3}}&&y_{3}b_{3}\\ 0&&\frac{x_{3}y_{3}}{x_{4}}\end{bmatrix}\begin{bmatrix}y_{2}&&0\\ \frac{b_{4}y_{2}}{x_{2}x_{3}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{b_{1}}{x_{1}x_{2}}&&y_{1}\end{bmatrix})
=tr[y2y3y4x1x3b2b4+y3y4x32b1b2+y2y3x12b3b4+y2x12x2x4+y3x1x3b1b3x1x2x3x4y1y3y4x3b2+y1y3x1b3x4y2y3y4x1b4+y3y4x3b1x12x2y1y3y4x3x1]\displaystyle=\text{tr}\begin{bmatrix}\frac{y_{2}y_{3}y_{4}x_{1}x_{3}b_{2}b_{4}+y_{3}y_{4}x_{3}^{2}b_{1}b_{2}+y_{2}y_{3}x_{1}^{2}b_{3}b_{4}+y_{2}x_{1}^{2}x_{2}x_{4}+y_{3}x_{1}x_{3}b_{1}b_{3}}{x_{1}x_{2}x_{3}x_{4}}&&\frac{y_{1}y_{3}y_{4}x_{3}b_{2}+y_{1}y_{3}x_{1}b_{3}}{x_{4}}\\ \frac{y_{2}y_{3}y_{4}x_{1}b_{4}+y_{3}y_{4}x_{3}b_{1}}{x_{1}^{2}x_{2}}&&\frac{y_{1}y_{3}y_{4}x_{3}}{x_{1}}\end{bmatrix}
=y1y3y4x2x32x4+y2y3y4x1x3b2b4+y3y4x32b1b2+y2y3x12b3b4+y2x12x2x4+y3x1x3b1b3x1x2x3x4\displaystyle=\frac{y_{1}y_{3}y_{4}x_{2}x_{3}^{2}x_{4}+y_{2}y_{3}y_{4}x_{1}x_{3}b_{2}b_{4}+y_{3}y_{4}x_{3}^{2}b_{1}b_{2}+y_{2}y_{3}x_{1}^{2}b_{3}b_{4}+y_{2}x_{1}^{2}x_{2}x_{4}+y_{3}x_{1}x_{3}b_{1}b_{3}}{x_{1}x_{2}x_{3}x_{4}}

Letting b1=b2=b3=b4=1b_{1}=b_{2}=b_{3}=b_{4}=1 and rearranging the terms in our numerator, we conclude

χγ,T=Px(P)yT(P)crossT(γ)=y2x12x2x4+y2y3x12+(y3+y2y3y4)x1x3+y3y4x32+y1y3y4x2x32x4x1x2x3x4.\chi_{\gamma,\mathcal{L}_{T}^{\prime}}=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}^{\prime}}(P)}{\text{cross}_{T}(\gamma)}=\frac{y_{2}x_{1}^{2}x_{2}x_{4}+y_{2}y_{3}x_{1}^{2}+(y_{3}+y_{2}y_{3}y_{4})x_{1}x_{3}+y_{3}y_{4}x_{3}^{2}+y_{1}y_{3}y_{4}x_{2}x_{3}^{2}x_{4}}{x_{1}x_{2}x_{3}x_{4}}.
Remark 5.18 (Comparision to Example 3.23 in [8]).

We chose to revisit Example 3.23 from [8] using a different principal lamination to demonstrate the differences in our construction. In [8], they use the same triangulation and closed curve with the principal lamination T\mathcal{L}_{T} defined by bT(T,τi)=1b_{T}(\mathcal{L}_{T},\tau_{i})=1 for each 1i41\leq i\leq 4. For our example, we consider T\mathcal{L}_{T}^{\prime} defined by taking T\mathcal{L}_{T} and changing bT(T,τ2)b_{T}(\mathcal{L}_{T},\tau_{2}) from 11 to 1-1 and keeping everything else the same.

With these differences, Musiker and Williams obtain

χγ,T=Px(P)yT(P)crossT(γ)=x12x2x4+y3x12+(y2y3+y3y4)x1x3+y2y3y4x32+y1y2y3y4x2x32x4x1x2x3x4,\chi_{\gamma,\mathcal{L}_{T}}=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\gamma)}=\frac{x_{1}^{2}x_{2}x_{4}+y_{3}x_{1}^{2}+(y_{2}y_{3}+y_{3}y_{4})x_{1}x_{3}+y_{2}y_{3}y_{4}x_{3}^{2}+y_{1}y_{2}y_{3}y_{4}x_{2}x_{3}^{2}x_{4}}{x_{1}x_{2}x_{3}x_{4}},

whereas, we have

χγ,T=Px(P)yT(P)crossT(γ)==y2x12x2x4+y2y3x12+(y3+y2y3y4)x1x3+y3y4x32+y1y3y4x2x32x4x1x2x3x4.\chi_{\gamma,\mathcal{L^{\prime}}_{T}}=\frac{\sum_{P}x(P)y_{\mathcal{L^{\prime}}_{T}}(P)}{\text{cross}_{T}(\gamma)}==\frac{y_{2}x_{1}^{2}x_{2}x_{4}+y_{2}y_{3}x_{1}^{2}+(y_{3}+y_{2}y_{3}y_{4})x_{1}x_{3}+y_{3}y_{4}x_{3}^{2}+y_{1}y_{3}y_{4}x_{2}x_{3}^{2}x_{4}}{x_{1}x_{2}x_{3}x_{4}}.
Remark 5.19.

Also, notice that Figure 8 is arranged to look like the Hasse diagram for a poset. Although we may always arrange the set of (good) perfect matchings as a poset with the cover relation given by “flipping” local tiles as in [7], the coefficient variables cannot be directly recovered without the lamination. For instance, the minimal element in this poset has a non-trivial height monomial of y2y_{2}. This is not typically how the yy-coefficients or height monomial is classically defined. In general, there will exist a choice of isotopic representative of arc that gives the classical poset structure interpretation of the coefficients, but perturbations of this arc will change the poset structure. An example of this can be seen comparing the poset we obtain in Figure 15 and Figure 12 in Section 6.

6. Expansion formula for one-sided closed curves using matrix products

In this section, we will prove the matrix formulae in the previous section can be used to find the Laurent expansion for one-sided closed curves. To prove this result, we will be following the general guidelines found in Section 5 of this paper. That is, we will be associating 2×22\times 2 matrices to the parallelograms of a band graph, which will then give us a way of representing the good matching enumerator of our graph in terms of the trace of a product of matrices. To conclude this section, we will show that the matrix product associated with the band graphs of one-sided curves coincides with the matrix product associated to a canonical MM-path associated to the one-sided curve.

6.1. Good matching enumerators for one-sided closed curves.

The basic ingredients used to construct band graphs for one-sided curves are largely the same as those found in Definition 5.10, but there is a slight twist.

For two-sided closed curves, we always glue together sides with the same sign, but when working with the one-sided closed curve, we will now be identifying edges that have the opposite sign. With this in mind, the following abstract band graph will differ from the one defined earlier.

Definition 6.1 (Abstract band graph for one-sided curves).

An abstract band graph for a one-sided curve with dd tiles is formed by concatenating the following puzzle pieces:

  • An initial triangle

    a\displaystyle ax\displaystyle xy\displaystyle yi1\displaystyle i_{1}id\displaystyle i_{d}
  • d1d-1 parallelograms H1,,Hd1H_{1},\ldots,H_{d-1}, where each HjH_{j} is as before.

  • A final triangle based on whether dd is odd or even.

    id\displaystyle i_{d}i1\displaystyle i_{1}a\displaystyle a^{\prime}id\displaystyle i_{d}a\displaystyle a^{\prime}i1\displaystyle i_{1}EvenOddx\displaystyle x^{\prime}y\displaystyle y^{\prime}x\displaystyle x^{\prime}y\displaystyle y^{\prime}

Just like the orientable case, we associate various matrices to the parallelograms of our graph that will be determined by comparing the shape of the parallelograms HiH_{i} and Hi1H_{i-1}, as well as the principal lamination used for coefficients. See Definition 5.11.

We now provide a description of the entries of MdM_{d} in terms of weight and coefficient monomials x(P)x(P) and yT(P)y_{\mathcal{L}_{T}}(P) for good matchings of our new variant of abstract band graphs. The following proposition is an immediate corollary of Proposition 5.13 after relabeling the edges of our graph.

Proposition 6.2.

Let GG be an abstract snake graph with dd-tiles, but with a labeling obtained by substituting ww with aa^{\prime}, bb with idi_{d} and zz with i1i_{1}. Write

Md=[AdBdCdDd].M_{d}=\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}.

For d2d\geq 2,

  • When bT(T,id1)=1b_{T}(\mathcal{L}_{T},{i_{d-1}})=1 we have

    Ad=PSAx(P)yT(P)(xi1xid1)xaxa\displaystyle A_{d}=\frac{\sum_{P\in S_{A}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{1}}\cdots x_{i_{d-1}})x_{a}x_{a^{\prime}}} Bd=PSBx(P)yT(P)(xi2xid1)xidxa\displaystyle\qquad B_{d}=\frac{\sum_{P\in S_{B}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d-1}})x_{i_{d}}x_{a^{\prime}}}
    Cd=PSCx(P)yT(P)(xixid)xaxi1yid\displaystyle C_{d}=\frac{\sum_{P\in S_{C}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i}\cdots x_{i_{d}})x_{a}x_{i_{1}}y_{i_{d}}} Dd=PSDx(P)yT(P)(xi2xid)xidxi1yid\displaystyle\qquad D_{d}=\frac{\sum_{P\in S_{D}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d}})x_{i_{d}}x_{i_{1}}y_{i_{d}}}
  • When bT(T,id1)=1b_{T}(\mathcal{L}_{T},i_{d-1})=-1 we have

    Ad=PSAx(P)yT(P)(xi1xid1)xaxayid\displaystyle A_{d}=\frac{\sum_{P\in S_{A}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{1}}\cdots x_{i_{d-1}})x_{a}x_{a}^{\prime}y_{i_{d}}} Bd=PSBx(P)yT(P)(xi2xid1)xidxayid\displaystyle\qquad B_{d}=\frac{\sum_{P\in S_{B}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d-1}})x_{i_{d}}x_{a^{\prime}}y_{i_{d}}}
    Cd=PSCx(P)yT(P)(xixid)xaxi1\displaystyle C_{d}=\frac{\sum_{P\in S_{C}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i}\cdots x_{i_{d}})x_{a}x_{i_{1}}} Dd=PSDx(P)yT(P)(xi2xid)xidxi1\displaystyle\qquad D_{d}=\frac{\sum_{P\in S_{D}}x(P)y_{\mathcal{L}_{T}}(P)}{(x_{i_{2}}\cdots x_{i_{d}})x_{i_{d}}x_{i_{1}}}

For d=1d=1, the formulae remain the same, but the cases are determined by the sign of bT(T,i1)b_{T}(\mathcal{L}_{T},i_{1}).


Here, SA,SB,SCS_{A},S_{B},S_{C} and SDS_{D} are the sets of perfect matchings of GG which use the edges {a,a},{id,a},{a,i1}\{a,a^{\prime}\},\{i_{d},a^{\prime}\},\{a,i_{1}\} and {id,i1}\{i_{d},i_{1}\}, respectively.

The following comes as a direct corollary of Proposition 6.2.

Corollary 6.3.

Let GG be an abstract band graph corresponding to a one-sided closed curve with dd tiles. Write Md=[AdBdCdDd]M_{d}=\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}.

  • When bT(T,id)=1b_{T}({\mathcal{L}_{T}},i_{d})=1, then

    Px(P)y(P)xi1xid\displaystyle\frac{\sum_{P}x(P)y(P)}{x_{i_{1}}\cdots x_{i_{d}}} =xaAdxid+Bdxi1+xi1yidCd\displaystyle=\frac{x_{a}A_{d}}{x_{i_{d}}}+\frac{B_{d}}{x_{i_{1}}}+x_{i_{1}}y_{i_{d}}C_{d}
    =tr([xaxidxi1yid1xi10][AdBdCdDd])\displaystyle=\text{tr}(\begin{bmatrix}\frac{x_{a}}{x_{i_{d}}}&&x_{i_{1}}y_{i_{d}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}) (6.1)
  • When bT(T,id)=1b_{T}({\mathcal{L}_{T}},i_{d})=-1, then

    Px(P)y(P)xi1xid\displaystyle\frac{\sum_{P}x(P)y(P)}{x_{i_{1}}\cdots x_{i_{d}}} =xayidAdxid+yidBdxi1+xi1Cd\displaystyle=\frac{x_{a}y_{i_{d}}A_{d}}{x_{i_{d}}}+\frac{y_{i_{d}}B_{d}}{x_{i_{1}}}+x_{i_{1}}C_{d}
    =tr([xayidxidxi1yidxi10][AdBdCdDd])\displaystyle=\text{tr}(\begin{bmatrix}\frac{x_{a}y_{i_{d}}}{x_{i_{d}}}&&x_{i_{1}}\\ \frac{y_{i_{d}}}{x_{i_{1}}}&&0\end{bmatrix}\begin{bmatrix}A_{d}&&B_{d}\\ C_{d}&&D_{d}\end{bmatrix}) (6.2)

where the sum is over all good matchings of GG.

Proof.

Following the notation from Proposition 6.2, we consider the sets SA,SB,SCS_{A},S_{B},S_{C} and SDS_{D}. If GG is our snake graph from Proposition 6.2 and G~\widetilde{G} is the band obtained from identifying aa and aa^{\prime}, then every perfect matching from SA,SBS_{A},S_{B} and SCS_{C} descends to a good matching of G~\widetilde{G} after removing either aa or aa^{\prime}. On the other hand, no perfect matching from SDS_{D} descends to a good matching of G~\widetilde{G}. As all good matchings of G~\widetilde{G} are obtained uniquely from perfect matchings in SA,SBS_{A},S_{B} or SCS_{C}, the result follows. ∎

The corollary above immediately proves the following theorem.

Theorem 6.4.

Suppose GG is an abstract band graph with dd tiles that corresponds to a one-sided closed curve. Then its good matching enumerator is given by

  • Px(P)y(P)=xi1xidtr([xaxidxi1yid1xi10]Md), when bT(T,id)=1\sum_{P}x(P)y(P)=x_{i_{1}}\ldots x_{i_{d}}\text{tr}(\begin{bmatrix}\frac{x_{a}}{x_{i_{d}}}&&x_{i_{1}}y_{i_{d}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}M_{d}),\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1

  • Px(P)y(P)=xi1xidtr([xayidxidxi1yidxi10]Md), when bT(T,id)=1\sum_{P}x(P)y(P)=x_{i_{1}}\ldots x_{i_{d}}\text{tr}(\begin{bmatrix}\frac{x_{a}y_{i_{d}}}{x_{i_{d}}}&&x_{i_{1}}\\ \frac{y_{i_{d}}}{x_{i_{1}}}&&0\end{bmatrix}M_{d}),\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1

where the sum is over all good matchings of GG.

6.2. The standard MM-path for a one-sided closed curve.

In this section, for any one-sided closed curve α\alpha, we will associate to it a standard MM-path ρα\rho_{\alpha} and show that the associated matrix formula will have the same form as Equations (6.1) and (6.2) found in Corollary 6.3.

The elementary steps used in the standard MM-path will be the same as those used in the orientable case; however, we will need to make an adjustment to the matrix corresponding to the type III step that travels through the crosscap, as we will now be traveling along an arc from a point of the form vm,τ+v_{m,\tau}^{+} (vm,τv_{m,\tau}^{-}) to another point of the form vm,τ+v_{m^{\prime},\tau}^{+} (vm,τv_{m^{\prime},\tau}^{-}).

Definition 6.5 (Elementary Step of Type 3’).

For this variation of the third type of elementary step, we travel through the crosscap along a path parallel to a fixed arc τ\tau connecting two points vm,τ±v^{\pm}_{m,\tau} and vm,τ±v^{\pm}_{m^{\prime},\tau} associated to distinct marked points m,mm,m^{\prime}. The associated matrix is [0±xτ±1xτ0]\begin{bmatrix}0&\pm x_{\tau}\\ \pm\frac{1}{x_{\tau}}&0\end{bmatrix}, where we use xr,1xrx_{r},\frac{1}{x_{r}} if this step sees τ\tau on the right and xr,1xr-x_{r},\frac{-1}{x_{r}} otherwise.

hm\displaystyle h_{m}hm\displaystyle h_{m^{\prime}}m\displaystyle mm\displaystyle m^{\prime}τ\displaystyle\tauτ\displaystyle\tauvm,τ+\displaystyle v_{m,\tau}^{+}vm,τ+\displaystyle v_{m^{\prime},\tau}^{+}[0xτ1xτ0]\displaystyle\begin{bmatrix}0&x_{\tau}\\ \frac{1}{x_{\tau}}&0\end{bmatrix}hm\displaystyle h_{m}hm\displaystyle h_{m^{\prime}}m\displaystyle mm\displaystyle m^{\prime}τ\displaystyle\tauτ\displaystyle\tauvm,τ\displaystyle v_{m,\tau}^{-}vm,τ\displaystyle v_{m^{\prime},\tau}^{-}[0xτ1xτ0]\displaystyle\begin{bmatrix}0&-x_{\tau}\\ \frac{-1}{x_{\tau}}&0\end{bmatrix}
Figure 9. Elementary steps of type 3’ in the positive and negative direction

An illustration of the type 3’ elementary step can be found in Figure 10 of Section 6.3. The next lemma is a quick check verification that Lemma 5.6 still holds with our new type three step.

Lemma 6.6.

Fix (𝐒,𝐌)(\mathbf{S},\mathbf{M}) and TT. Let α\alpha be a one-sided closed curve. Then for any two MM-paths, ρ\rho and ρ\rho^{\prime} associated to α\alpha, we have

|tr(M(ρ))|=|tr(M(ρ))|.|\text{tr}(M(\rho))|=|\text{tr}(M(\rho^{\prime}))|.
Proof.

First, we observe if we have an MM-path M(ρ)=M(ρn)M(ρ1)M(\rho)=M(\rho_{n})\cdot\ldots\cdot M(\rho_{1}), then

tr[M(ρn)M(ρ1)]=tr[M(ρi)M(ρ1)M(ρn)M(ρi+1)]\text{tr}[M(\rho_{n})\cdot\ldots\cdot M(\rho_{1})]=\text{tr}[M(\rho_{i})\cdot\ldots M(\rho_{1})\cdot M(\rho_{n})\cdot\ldots\cdot M(\rho_{i+1})]

for all 1in1\leq i\leq n, so the trace is invariant under the starting point of our MM-path. The original three elementary steps are covered already by Lemma 5.6.

Furthermore, we have the equality

[0xτ1xτ0][yτ001]=[100yτ][0xτ1xτ0]\begin{bmatrix}0&&x_{\tau}\\ \frac{1}{x_{\tau}}&&0\end{bmatrix}\begin{bmatrix}y_{\tau}&&0\\ 0&&1\end{bmatrix}=\begin{bmatrix}1&&0\\ 0&&y_{\tau}\end{bmatrix}\begin{bmatrix}0&&x_{\tau}\\ \frac{1}{x_{\tau}}&&0\end{bmatrix}

which implies, with respect to |tr(M(ρ))||\text{tr}(M(\rho))|, crossing τ\tau with a step two move and then passing through the crosscap with the adjusted step three move is the same as first passing through the crosscap and then crossing τ\tau. ∎

Despite working on a non-orientable surface, the algorithm for constructing the standard MM-path for the one-sided closed curve will be reminiscent to the algorithm for creating standard MM-paths for closed curves on orientable surfaces. The differences being that we require the first arc to be counter-clockwise from the last; and we require that the final step of the path corresponds to traveling through the crosscap.

Definition 6.7.

(Standard MM-path for a one-sided closed curve) Let α\alpha be a one-sided closed curve which crosses dd arcs of a fixed triangulation TT (counted with multiplicity). Choose a triangle Δ\Delta in TT such that two of its arcs are crossed by α\alpha. We label these two arcs as τi1\tau_{i_{1}} and τid\tau_{i_{d}}, where τi1\tau_{i_{1}} is in the counterclockwise direction from τid\tau_{i_{d}}. We label the third side with aa. Let pp be a point on α\alpha which lies in Δ\Delta and has the form v,τi1±v_{*,\tau_{i_{1}}}^{\pm}. Letting, τi1,,τid\tau_{i_{1}},\ldots,\tau_{i_{d}} denote the ordered sequence of arcs that are crossed by α\alpha as we move from pp and Δ\Delta, we have that the standard MM-path ρα\rho_{\alpha} associated to α\alpha is the same as Definition 5.17, starting and ending at pp and traveling along elementary steps based on whether τij+1\tau_{i_{j+1}} is counterclockwise or clockwise from τij\tau_{i_{j}}. The final elementary steps of ρα\rho_{\alpha} will be determined by the fact that τi1\tau_{i_{1}} is in the counterclockwise direction from τid\tau_{i_{d}} and that, to return to pp, we must travel parallel to i1i_{1} through the crosscap. We consider indices modulo nn. See Figure 10 for an example of a such a standard MM-path.

We’re ready to prove the main theorem of this section.

Theorem 6.8.

Let (𝐒,𝐌)(\mathbf{S},\mathbf{M}) be a marked non-orientable surface with a triangulation TT, and let T={τi1,,τin}T=\{\tau_{i_{1}},\ldots,\tau_{i_{n}}\} be the corresponding triangulation. Let 𝒜T(𝐒,𝐌)\mathcal{A}_{\mathcal{L}_{T}}(\mathbf{S},\mathbf{M}) be the corresponding cluster algebra associated to the principal lamination T\mathcal{L}_{T}.

Suppose α\alpha is a one-sided closed curve which is not contractible, and has no contractible kinks. Then

χα,𝒯=1crossT(α)Px(P)yLT(P)\displaystyle\chi_{\alpha,\mathcal{L_{T}}}=\frac{1}{\text{cross}_{T}(\alpha)}\sum_{P}x(P)y_{L_{T}}(P)

where the sum is over all good matchings PP of the band graph 𝒢~T,α\widetilde{\mathcal{G}}_{T,\alpha}.

Proof.

We consider the standard MM-path of α\alpha defined in Definition 6.7. For the indices jj from τ1\tau_{1} to τd1\tau_{d-1}, we have the following four cases that depend on the sign of bT(T,τij)b_{T}(\mathcal{L}_{T},\tau_{i_{j}}) and whether τij+1\tau_{i_{j+1}} lies clockwise or counterclockwise from τij\tau_{i_{j}}.

[10xajxijxij+11][100yij]=[10xajxijxij+1yij](when bT(T,ij)=1,CCW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{a_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ 0&&y_{i_{j}}\end{bmatrix}=\begin{bmatrix}1&&0\\ \frac{x_{a_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&y_{i_{j}}\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=1,\text{CCW}) (6.3)
[10xajxijxij+11][yij001]=[yij0xajyijxijxij+11](when bT(T,ij)=1,CCW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{a_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&1\end{bmatrix}\begin{bmatrix}y_{i_{j}}&&0\\ 0&&1\end{bmatrix}=\begin{bmatrix}y_{i_{j}}&&0\\ \frac{x_{a_{j}}y_{i_{j}}}{x_{i_{j}}x_{i_{j+1}}}&&1\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=-1,\text{CCW}) (6.4)
[10xijxajxij+11][0xaj1xaj0][10xij+1xajxij1][100yij]=[xij+1xijxajyij0xijyijxij+1](when bT(T,ij)=1,CW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{i_{j}}}{x_{a_{j}}x_{i_{j+1}}}&&1\\ \end{bmatrix}\begin{bmatrix}0&&x_{a_{j}}\\ -\frac{1}{x_{a_{j}}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{i_{j+1}}}{x_{a_{j}}x_{i_{j}}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ 0&&y_{i_{j}}\end{bmatrix}=\begin{bmatrix}\frac{x_{i_{j+1}}}{x_{i_{j}}}&&x_{a_{j}}y_{i_{j}}\\ 0&&\frac{x_{i_{j}}y_{i_{j}}}{x_{i_{j+1}}}\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=1,\text{CW}) (6.5)
[10xijxajxij+11][0xaj1xaj0][10xij+1xajxij1][yij001]=[xij+1yijxijxaj0xijxij+1](when bT(T,ij)=1,CW)\displaystyle\begin{bmatrix}1&&0\\ \frac{x_{i_{j}}}{x_{a_{j}}x_{i_{j+1}}}&&1\\ \end{bmatrix}\begin{bmatrix}0&&x_{a_{j}}\\ -\frac{1}{x_{a_{j}}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{i_{j+1}}}{x_{a_{j}}x_{i_{j}}}&&1\end{bmatrix}\begin{bmatrix}y_{i_{j}}&&0\\ 0&&1\end{bmatrix}=\begin{bmatrix}\frac{x_{i_{j+1}}y_{i_{j}}}{x_{i_{j}}}&&x_{a_{j}}\\ 0&&\frac{x_{i_{j}}}{x_{i_{j+1}}}\end{bmatrix}(\text{when }b_{T}(\mathcal{L}_{T},i_{j})=-1,\text{CW}) (6.6)

Here, aja_{j} is the third side of the triangle with sides τij,τij+1\tau_{i_{j}},\tau_{i_{j+1}}.

As τi1\tau_{i_{1}} is in the counterclockwise direction from τid\tau_{i_{d}}, and we must travel parallel along τi1\tau_{i_{1}} through the crosscap in order to return to pp, the final few steps of ρα\rho_{\alpha} are represented by one of the two matrices:

[0xi11xi10][10xaxi1xidyid]=[xaxidxi1yid1xi10] when bT(T,id)=1.\displaystyle\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{a}}{x_{i_{1}}x_{i_{d}}}&&y_{i_{d}}\end{bmatrix}=\begin{bmatrix}\frac{x_{a}}{x_{i_{d}}}&&x_{i_{1}}y_{i_{d}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1. (6.7)
[0xi11xi10][yid0xayidxi1xid1]=[xayidxidxi1yidxi10] when bT(T,id)=1.\displaystyle\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\begin{bmatrix}y_{i_{d}}&&0\\ \frac{x_{a}y_{i_{d}}}{x_{i_{1}}x_{i_{d}}}&&1\end{bmatrix}=\begin{bmatrix}\frac{x_{a}y_{i_{d}}}{x_{i_{d}}}&&x_{i_{1}}\\ \frac{y{{}_{i_{d}}}}{x_{i_{1}}}&&0\end{bmatrix}\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1. (6.8)

By Definition 5.7, Theorem 6.4 and using the interpretation of χα,T\chi_{\alpha,\mathcal{L}_{T}} in terms of good matchings on some abstract band graph 𝒢\mathcal{G} we have

  • χα,T=tr([xaxidxi1yid1xi10]M)=Px(P)yT(P)crossT(α), when bT(T,id)=1\chi_{\alpha,\mathcal{L}_{T}}=\text{tr}\left(\begin{bmatrix}\frac{x_{a}}{x_{i_{d}}}&&x_{i_{1}}y_{i_{d}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}M\right)=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\alpha)},\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=1

  • χα,T=tr([xayidxidxi1yidxi10]M)=Px(P)yT(P)crossT(α), when bT(T,id)=1\chi_{\alpha,\mathcal{L}_{T}}=\text{tr}\left(\begin{bmatrix}\frac{x_{a}y_{i_{d}}}{x_{i_{d}}}&&x_{i_{1}}\\ \frac{y_{i_{d}}}{x_{i_{1}}}&&0\end{bmatrix}M\right)=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\alpha)},\text{ when }b_{T}(\mathcal{L}_{T},i_{d})=-1

where MM is a obtained by multiplying together a sequence of matrices of the four forms mentioned forms (6.3), (6.4), (6.5), (6.6) and can be interpreted as the matrix MdM_{d} for 𝒢¯T,γ\overline{\mathcal{G}}_{T,\gamma} by comparing with Definition 5.11. This completes the proof. ∎

If ρ\rho is a non-standard MM-path for the one-sided closed curve α\alpha, then ρ\rho can be deformed into a standard MM-path ρα\rho_{\alpha} by the local adjustments found in Lemma 4.8 of [8] and Lemma 6.6.

Lemma 6.9.

If we use the standard MM-path, Definition 6.7, for a one-sided closed curve α\alpha with no contractible kinks, then every coefficient of χα,T\chi_{\alpha,\mathcal{L}_{T}} is positive.

Proof.

Given a one-sided closed curve α\alpha with no contractible kinks, let ρα\rho_{\alpha} denote the corresponding standard MM-path. The proof of this lemma comes from the simple observation that all entries of matrices of the forms (6.3), (6.4), (6.5), (6.6), (6.7) and (6.8) are positive. As such, χα,T\chi_{\alpha,\mathcal{L}_{T}} is a sum of positive terms and is thus positive. ∎

An immediate corollary of Theorem 6.8 and Lemma 6.9 is the following.

Corollary 6.10.

The quantity χα,T\chi_{\alpha,\mathcal{L}_{T}} is a Laurent polynomial with all coefficients positive. This verifies and provides another proof of the positivity result by Wilson in [11].

Next, we need to make sure that if we fix a principal lamination T\mathcal{L}_{T}, then the isotopic representations of a one-sided curve α\alpha share the same Laurent polynomial. That is, if α\alpha and α\alpha^{\prime} are isotopic one-sided closed curves, then

χα,T=χα,T.\chi_{\alpha,\mathcal{L}_{T}}=\chi_{\alpha^{\prime},\mathcal{L}_{T}}.

Throughout this segment, we will write

χα,T=Px(P)yT(P)crossT(α) and χα,T=Px(P)yT(P)crossT(α)\chi_{\alpha,\mathcal{L}_{T}}=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\alpha)}\qquad\text{ and }\qquad\chi_{\alpha^{\prime},\mathcal{L}_{T}}=\frac{\sum_{P}x^{\prime}(P)y_{\mathcal{L}_{T}}^{\prime}(P)}{\text{cross}_{T}(\alpha^{\prime})}

While this statement may seem straightforward, there is some difficulty that occurs due to the fact that, in most situations, the band graphs of α\alpha and α\alpha^{\prime} are not isomorphic to one another and hence yield different poset structures. This is illustrated in Figure 18.

Lemma 6.11.

Let α\alpha be a one-sided closed curve and T\mathcal{L}_{T} be a principal lamination on the triangulation TT. Let α\alpha^{\prime} be the reflection of α\alpha about the crosscap. Then

χα,T=χα,T.\chi_{\alpha,\mathcal{L}_{T}}=\chi_{\alpha^{\prime},\mathcal{L}_{T}}.
Proof.

By definition, we have crossT(α)=crossT(α)\text{cross}_{T}(\alpha)=\text{cross}_{T}(\alpha^{\prime}); furthermore, one can obtain the band graph 𝒢~T,α\widetilde{\mathcal{G}}_{T,\alpha^{\prime}} of α\alpha^{\prime} by taking 𝒢~T,α\widetilde{\mathcal{G}}_{T,\alpha} and reversing the relative orientation of each tile. This ensures that the set of perfect matchings on both graphs are the same, and x(P)=x(P)x(P)=x^{\prime}(P) for each perfect matching PP. Note, the minimal matching on 𝒢~T,α\widetilde{\mathcal{G}}_{T,\alpha} is the maximum matching on 𝒢~T,α\widetilde{\mathcal{G}}_{T,\alpha^{\prime}} (and vice versa); however, reflecting α\alpha across the crosscap has the effect of changing SS-intersections into ZZ-intersections (and vice versa).

As a corollary, yT(P)=yT(P)y_{\mathcal{L}_{T}}(P)=y_{\mathcal{L}_{T}}^{\prime}(P) for this specific matching. The remaining matchings on both band graphs are found by taking PP, flipping the orientation of certain diagonals and repeating this process until conclusion. We thus have

χα,T=Px(P)yT(P)crossT(α)=Px(P)yT(P)crossT(α)=χα,T.\chi_{\alpha,\mathcal{L}_{T}}=\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\alpha)}=\frac{\sum_{P}x^{\prime}(P)y_{\mathcal{L}_{T}}^{\prime}(P)}{\text{cross}_{T}(\alpha^{\prime})}=\chi_{\alpha^{\prime},\mathcal{L}_{T}}.

Proposition 6.12.

Let α\alpha and α\alpha^{\prime} be isotopic one-sided closed curves. Then

χα,T=χα,T.\chi_{\alpha,\mathcal{L}_{T}}=\chi_{\alpha^{\prime},\mathcal{L}_{T}}.
Proof.

We begin by setting up some notation. Let τ1,,τn\tau_{1},\ldots,\tau_{n} be the arcs of our triangulation TT and let T\mathcal{L}_{T} be a principal lamination. As a reminder, each τi\tau_{i} lifts to two arcs in the orientable double cover that will have opposite signs with respect to the principal lamination. Let τi¯\overline{\tau_{i}} be the copy of τi\tau_{i} that lifts to an SS-shape intersection and τi~\widetilde{\tau_{i}} be the copy that lifts to a ZZ-shape intersection.

To prove this result, we show χ\chi is invariant under “rotations” of the one-sided closed curve around the crosscap. In other words, if τi\tau_{i} and τi+1\tau_{i+1} are the first two arcs that α\alpha intersects and τi1\tau_{i-1} is the final arc α\alpha intersects, then we will assume α\alpha^{\prime} intersects τi+1\tau_{i+1} first and τi\tau_{i} last (see Figure 17) and show χα,T=χα,T\chi_{\alpha,\mathcal{L}_{T}}=\chi_{\alpha^{\prime},\mathcal{L}_{T}}. Note that the τi\tau_{i}’s in α\alpha and α\alpha^{\prime} correspond to arcs on the orientable surface with opposite sign with respect to T\mathcal{L}_{T}. By Lemma 6.11, we may assume without loss of generality, that τi+1\tau_{i+1} lies counterclockwise to τi\tau_{i}. This implies that on the other side of the crosscap, τi+1\tau_{i+1} lies clockwise to τi\tau_{i} (otherwise, we can reflect across the crosscap). Further, we will assume the τi\tau_{i} that α\alpha crosses lifts to τi¯\overline{\tau_{i}}, while the τi\tau_{i} that α\alpha^{\prime} crosses lifts to τi~\widetilde{\tau_{i}}.

By Theorem 6.8 and the fact tr(UV)=tr(VU)\text{tr}(UV)=\text{tr}(VU) for two matrices UU and VV, we have

χα,T\displaystyle\chi_{\alpha,\mathcal{L}_{T}} =tr([0xi1xi0]M[10xaixixi+11][100yi])\displaystyle=\text{tr}(\begin{bmatrix}0&&x_{i}\\ \frac{1}{x_{i}}&&0\end{bmatrix}\cdot M\cdot\begin{bmatrix}1&&0\\ \frac{x_{a_{i}}}{x_{i}x_{i+1}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ 0&&y_{i}\end{bmatrix})
=tr([10xaixixi+11][100yi][0xi1xi0]M)\displaystyle=\text{tr}(\begin{bmatrix}1&&0\\ \frac{x_{a_{i}}}{x_{i}x_{i+1}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ 0&&y_{i}\end{bmatrix}\begin{bmatrix}0&&x_{i}\\ \frac{1}{x_{i}}&&0\end{bmatrix}\cdot M)
=tr([0xiyixixaixi+1]M)\displaystyle=\text{tr}(\begin{bmatrix}0&&x_{i}\\ \frac{y_{i}}{x_{i}}&&\frac{x_{a_{i}}}{x_{i+1}}\end{bmatrix}\cdot M)
=tr([0xi+11xi+11][10xixaixi+11][0xai1xai0][10xi+1xaixi1][yi001]M)\displaystyle=\text{tr}(\begin{bmatrix}0&&x_{i+1}\\ \frac{1}{x_{i+1}}&&1\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{i}}{x_{a_{i}}x_{i+1}}&&1\end{bmatrix}\begin{bmatrix}0&&x_{a_{i}}\\ -\frac{1}{x_{a_{i}}}&&0\end{bmatrix}\begin{bmatrix}1&&0\\ \frac{x_{i+1}}{x_{a_{i}}x_{i}}&&1\end{bmatrix}\begin{bmatrix}y_{i}&&0\\ 0&&1\end{bmatrix}\cdot M)
=χα,T\displaystyle=\chi_{\alpha^{\prime},\mathcal{L}_{T}}

where MM is a product of matrices determined by the standard MM-path of both α\alpha and α\alpha^{\prime}, i.e. it is a product of matrices (6.3), (6.4), (6.5), (6.6). The above proof holds, even if we were to switch the signs of τi¯\overline{\tau_{i}} and τi~\widetilde{\tau_{i}} with respect to T\mathcal{L}_{T}. ∎

Remark 6.13.

One could also prove this by making use of the band graphs of these curves. Namely, a rotation of the α\alpha amounts to moving the first tile of the graph to the end of the graph and then flipping the sign of the lamination. This method is far more difficult to prove in generality than the matrix method used above.

6.3. Möbius Band Example

In this subsection, we illustrate the techniques used above in a few examples involving the Möbius Band. We will first compute the Laurent expansion by computing the trace of the MM-path matrix and then we will compare it to the Laurent expansion obtained from the good matching enumerator.

Notation 6.14.

Throughout this section, if we have an arc xx in the triangulation, we let XX denote the coefficient yxy_{x} corresponding to xx.

1234zwyxabcdρ1\displaystyle\rho_{1}ρ13\displaystyle\rho_{13}ρ2\displaystyle\rho_{2}α\displaystyle\alpha\displaystyle\dotsc
Figure 10. The one-sided curve α\alpha on M4M_{4} along with the standard MM-path which begins with ρ1\rho_{1} and ends with ρ13\rho_{13}.

The one-sided curve α\alpha on M4M_{4} and each step of the standard MM-path are shown in Figure 10. We assume each arc w,x,y,zw,x,y,z that α\alpha intersects lifts to an SS-intersection in the orientable double cover and will look at what happens on different laminations later. The matrices associated to each of step of the standard MM-path are listed here:

M(ρ1)=[100Z],M(ρ2)=[10cwz1],M(ρ3)=[100W],M(ρ4)=[10xaw1],M(ρ5)=[0a1a0],M(ρ6)=[10wax1],M(ρ7)=[100X],M(ρ8)=[10yxb1],M(ρ9)=[0b1b0],M(ρ10)=[10xby1],M(ρ11)=[100Y],M(ρ12)=[10dyz1],\displaystyle\begin{array}[]{{>{\displaystyle}l}}M(\rho_{1})\ =\ \begin{bmatrix}1&0\\ 0&Z\end{bmatrix},\ M(\rho_{2})\ =\ \begin{bmatrix}1&0\\ \frac{c}{wz}&1\end{bmatrix},\ M(\rho_{3})\ =\ \begin{bmatrix}1&0\\ 0&W\end{bmatrix},\ M(\rho_{4})\ =\ \begin{bmatrix}1&0\\ \frac{x}{aw}&1\end{bmatrix},\\ \\ M(\rho_{5})\ =\ \begin{bmatrix}0&a\\ \frac{-1}{a}&0\end{bmatrix},\ M(\rho_{6})\ =\ \begin{bmatrix}1&0\\ \frac{w}{ax}&1\end{bmatrix},\ M(\rho_{7})\ =\ \begin{bmatrix}1&0\\ 0&X\end{bmatrix},\ M(\rho_{8})\ =\ \begin{bmatrix}1&0\\ \frac{y}{xb}&1\end{bmatrix},\\ \\ M(\rho_{9})\ =\ \begin{bmatrix}0&b\\ \frac{-1}{b}&0\end{bmatrix},\ M(\rho_{10})\ =\ \begin{bmatrix}1&0\\ \frac{x}{by}&1\end{bmatrix},\ M(\rho_{11})\ =\ \begin{bmatrix}1&0\\ 0&Y\end{bmatrix},\ M(\rho_{12})\ =\ \begin{bmatrix}1&0\\ \frac{d}{yz}&1\end{bmatrix},\\ \end{array}

M(ρ13)=[0z1z0]\displaystyle M(\rho_{13})\ =\ \begin{bmatrix}0&z\\ \frac{1}{z}&0\end{bmatrix}

Therefore, we have

M(ρα)=M(ρ13)[M(ρ12)M(ρ11)][M(ρ10)M(ρ9)M(ρ8)M(ρ7)][M(ρ6)M(ρ5)M(ρ4)M(ρ3)][M(ρ2)M(ρ1)]=[0z1z0][10dyzY][yxbX0xXy][xwaW0wWx][10cwzZ]=[cwxzWXY+bcdwWX+bcdwWX+acdyW+dxyzwxyzwxzWXYZ+bdwWXZ+adyWZxybcwWX+acyW+xyzwxz2bwWXZ+ayWZxz]Trace(M(ρα))=χα,T=cwxzWXY+bw2yWXZ+bcdwWX+awy2WZ+acdyW+dxyzwxyz\displaystyle\begin{array}[]{{>{\displaystyle}l}}\ M(\rho_{\alpha})=M(\rho_{13})\cdot[M(\rho_{12})\cdot M(\rho_{11})]\cdot[M(\rho_{10})\cdot M(\rho_{9})\cdot M(\rho_{8})\cdot M(\rho_{7})]\cdot[M(\rho_{6})\cdot M(\rho_{5})\cdot\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(\rho_{4})\cdot M(\rho_{3})]\cdot[M(\rho_{2})\cdot M(\rho_{1})]\\ \\ \ \ \ \ \ \ \ \ \ \ \ =\begin{bmatrix}0&z\\ \frac{1}{z}&0\end{bmatrix}\cdot\begin{bmatrix}1&0\\ \frac{d}{yz}&Y\end{bmatrix}\cdot\begin{bmatrix}\frac{y}{x}&bX\\ 0&\frac{xX}{y}\end{bmatrix}\cdot\begin{bmatrix}\frac{x}{w}&aW\\ 0&\frac{wW}{x}\end{bmatrix}\cdot\begin{bmatrix}1&0\\ \frac{c}{wz}&Z\end{bmatrix}\\ \\ \ \ \ \ \ \ \ \ \ \ \ =\begin{bmatrix}\frac{cwxzWXY+bcdwWX+bcdwWX+acdyW+dxyz}{wxyz}&\frac{wxzWXYZ+bdwWXZ+adyWZ}{xy}\\ &\\ \frac{bcwWX+acyW+xyz}{wxz^{2}}&\frac{bwWXZ+ayWZ}{xz}\end{bmatrix}\\ \\ \Rightarrow\mathbf{\text{Trace}}(M(\rho_{\alpha}))=\chi_{\alpha,\mathcal{L}_{T}}=\frac{cwxzWXY+bw^{2}yWXZ+bcdwWX+awy^{2}WZ+acdyW+dxyz}{wxyz}\\ \end{array}

where the closed brackets, in the M(ρα)M(\rho_{\alpha}) product, group the matrices by triangle and the final matrix, M(ρ13)M(\rho_{13}), corresponds to going through the crosscap.

Meanwhile, we have the band graph 𝒢¯α,T\overline{\mathcal{G}}_{\alpha,T} shown in Figure 11. With this band graph, the good matchings are demonstrated in Figure 12.

zwxyddwwyyzxxzcab4122333344
Figure 11. Band graph 𝒢~α,T\widetilde{\mathcal{G}}_{\alpha,T} for M4M_{4}.

Computing the good matching enumerator, we find

Px(P)yT(P)crossT(α)=cwxzWXY+bw2yWXZ+bcdwWX+awy2WZ+acdyW+dxyzwxyz\frac{\sum_{P}x(P)y_{\mathcal{L}_{T}}(P)}{\text{cross}_{T}(\alpha)}=\frac{cwxzWXY+bw^{2}yWXZ+bcdwWX+awy^{2}WZ+acdyW+dxyz}{wxyz}

and we see the two methods of computing the expansion agree with one another.

Refer to caption
Figure 12. Poset structure associated with 𝒢¯α,T\overline{\mathcal{G}}_{\alpha,T}.

6.4. Reflection and Rotation Example

If we instead considered the curve given by reflected α\alpha across the crosscap, which we will denote by α\alpha^{\prime}, then we get the MM-path found in Figure 13. The expansion is given by

χα,T\displaystyle\chi_{\alpha^{\prime},\mathcal{L}_{T}} =tr([0z1z0][zYyd0yz][X0bXxy1][W0aWwx1][wZzc0zw])\displaystyle=\text{tr}(\begin{bmatrix}0&&z\\ \frac{1}{z}&&0\end{bmatrix}\begin{bmatrix}\frac{zY}{y}&&d\\ 0&&\frac{y}{z}\end{bmatrix}\begin{bmatrix}X&&0\\ \frac{bX}{xy}&&1\end{bmatrix}\begin{bmatrix}W&&0\\ \frac{aW}{wx}&&1\end{bmatrix}\begin{bmatrix}\frac{wZ}{z}&&c\\ 0&&\frac{z}{w}\end{bmatrix})
=cwxzWXY+bw2yWXZ+bcdwWX+awy2WZ+acdyW+dxyzwxyz,\displaystyle=\frac{cwxzWXY+bw^{2}yWXZ+bcdwWX+awy^{2}WZ+acdyW+dxyz}{wxyz},

which agrees with out previous computation. Observe that when flipping across the crosscap, every counterclockwise sequence becomes clockwise and vice-versa. Additionally, the sign of the principal lamination changes as we are now intersecting arcs that lift to ZZ-intersections, so we must change each matrix of type two in the standard MM-path.

1234zwyxabcdρ1\displaystyle\rho_{1}ρ13\displaystyle\rho_{13}ρ2\displaystyle\rho_{2}\displaystyle\dotscα\displaystyle\alpha^{\prime}
Figure 13. MM-path after reflecting α\alpha from the previous example across the crosscap

The corresponding band graph is shown in Figure 14. Observe that it is the same as the band graph shown in Figure 11 with the exception that we have used the opposite relative orientation.

Observe that the minimal matching shown in Figure 12 (the bottommost graph of the figure), becomes the maximal matching for 𝒢~α,T\widetilde{\mathcal{G}}_{\alpha^{\prime},T} (see Figure 16); however, as we are now lifting to ZZ-intersections, the coefficient monomial remains the exact same as the previous example. A similar logic applies to the other good matchings on 𝒢~α,T\widetilde{\mathcal{G}}_{\alpha^{\prime},T} and the set of good matchings is shown to give comparison in Figure 15.

Next, we will look at a “rotation” of α\alpha. Let α\alpha be the arc depicted in Figure 10 and let β\beta be the one-sided closed curve shown in Figure 17. The matrices corresponding to the individual steps of the standard MM-path for β\beta are listed below. The first ten matrices appear in the matrix product for α\alpha as well, i.e, the matrices in purple text.

M(ρ1)=[100W],M(ρ2)=[10xaw1],M(ρ3)=[0a1a0],M(ρ4)=[10wax1],M(ρ5)=[100X],M(ρ6)=[10ybx1],M(ρ7)=[0b1b0],M(ρ8)=[10xby1],M(ρ9)=[100Y],M(ρ10)=[10dyz1],M(ρ11)=[Z001],M(ρ12)=[10wcz1],\displaystyle\begin{array}[]{{>{\displaystyle}l}}M(\rho_{1})\ =\ \begin{bmatrix}1&0\\ 0&W\end{bmatrix},\ M(\rho_{2})\ =\ \begin{bmatrix}1&0\\ \frac{x}{aw}&1\end{bmatrix},\ M(\rho_{3})\ =\ \begin{bmatrix}0&a\\ \frac{-1}{a}&0\end{bmatrix},\ M(\rho_{4})\ =\ \begin{bmatrix}1&0\\ \frac{w}{ax}&1\end{bmatrix},\\ \\ M(\rho_{5})\ =\ \begin{bmatrix}1&0\\ 0&X\end{bmatrix},\ M(\rho_{6})\ =\ \begin{bmatrix}1&0\\ \frac{y}{bx}&1\end{bmatrix},\ M(\rho_{7})\ =\ \begin{bmatrix}0&b\\ \frac{-1}{b}&0\end{bmatrix},\ M(\rho_{8})\ =\ \begin{bmatrix}1&0\\ \frac{x}{by}&1\end{bmatrix},\\ \\ M(\rho_{9})\ =\ \begin{bmatrix}1&0\\ 0&Y\end{bmatrix},\ M(\rho_{10})\ =\ \begin{bmatrix}1&0\\ \frac{d}{yz}&1\end{bmatrix}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},\ M(\rho_{11})\ =\ \begin{bmatrix}Z&0\\ 0&1\end{bmatrix},\ M(\rho_{12})\ =\ \begin{bmatrix}1&0\\ \frac{w}{cz}&1\end{bmatrix}\ ,\\ \end{array}

M(ρ13)=[0c1c0],M(ρ14)=[10zcw1],M(ρ15)=[0w1w0]\displaystyle M(\rho_{13})\ =\ \begin{bmatrix}0&c\\ \frac{-1}{c}&0\end{bmatrix},\ M(\rho_{14})\ =\ \begin{bmatrix}1&0\\ \frac{z}{cw}&1\end{bmatrix},\ M(\rho_{15})\ =\ \begin{bmatrix}0&w\\ \frac{1}{w}&0\end{bmatrix}

The proof of Proposition 6.12 shows that the trace of the matrix product associated to ρα\rho_{\alpha} agrees with the trace of the matrix product associated to ρβ\rho_{\beta}. This can be seen explicitly by computing the matrix product via Macaulay2. One finds

Trace(M(ρβ))\displaystyle\text{Trace}(M(\rho_{\beta})) =Trace(M(ρ15)M(ρ1))\displaystyle=\text{Trace}(M(\rho_{15})\cdots M(\rho_{1}))
=[dwwxzWXY+bdwWX+adyWxywyZ+cdw2zcwxzWXY+bw2yWXZ+bcdwWX+awy2WZ+acdyWwxyz]\displaystyle=\begin{bmatrix}\frac{d}{w}&&\frac{wxzWXY+bdwWX+adyW}{xy}\\ \frac{wyZ+cd}{w^{2}z}&&\frac{cwxzWXY+bw^{2}yWXZ+bcdwWX+awy^{2}WZ+acdyW}{wxyz}\end{bmatrix}
=cwxzWXY+bw2yWXZ+bcdwWX+awy2WZ+acdyW+dxyzwxyz.\displaystyle=\frac{cwxzWXY+bw^{2}yWXZ+bcdwWX+awy^{2}WZ+acdyW+dxyz}{wxyz}.

A similar computation can be done with the other rotations of α\alpha.


zwxyddwwyyzxxzcab4122333344
Figure 14. Band graph 𝒢~α,T\widetilde{\mathcal{G}}_{\alpha^{\prime},T}
zwxyzwxyzwxyzwxyzwxyzwxyWXZXZY
Figure 15. Perfect matchings on 𝒢~α,T\widetilde{\mathcal{G}}_{\alpha^{\prime},T} which creates a different poset structure than 𝒢~α,T\widetilde{\mathcal{G}}_{\alpha,T} using the flip definition from [7].
zwxyddwwyyzxxzcab4122333344
Figure 16. Maximal Matching on 𝒢~α,T\widetilde{\mathcal{G}}_{\alpha^{\prime},T}
1234zwyxabcdρ1\displaystyle\rho_{1}ρ13\displaystyle\rho_{13}ρ2\displaystyle\rho_{2}α\displaystyle\alpha\displaystyle\dotsc1234zwyxabcdβ\beta
Figure 17. Rotation of α\alpha to an isotopic one-sided closed curve β\beta

Refer to caption
Figure 18. Band graph of β\beta obtained by rotating α\alpha. Despite the band graphs being non-isomorphic, they provide the same expansion due to β\beta having a ZZ-intersection with zz

7. Skein Relations for One-Sided Closed Curves

To conclude this paper, we briefly show how the theory we have developed throughout this paper can be used to prove skein relations on non-orientable surfaces. Skein relations on non-orientable surfaces have already been proven for xx-coordinates in Section 4.3 of [1]; however, with our matrix formulae, we can show these equations hold true when considering coefficients that come from principal laminations.

There are only a few cases that we need to consider, and most follow from basic adjustments to the arguments found in [8] using Sections 5 and 6 of this paper. Throughout this section, we will let e(γ,τ)e(\gamma,\tau) denote the number of crossings between the generalized arc/loop/one-sided closed curve γ\gamma and the arc τ\tau.

Case One: Intersection of Generalized Arcs

In this case, we do not consider one-sided closed curves, so we can simply lift to the orientable double cover and apply Proposition 6.4 from [8]. Considering general principal laminations rather than principal coefficients, the statement becomes the following.

Proposition 7.1.

Let γ1\gamma_{1} and γ2\gamma_{2} be two generalized arcs which intersect each other at least once; let xx be a point of intersection; and let α1,α2\alpha_{1},\alpha_{2} and β1,β2\beta_{1},\beta_{2} be the two pairs of arcs obtained by resolving the intersection of γ1\gamma_{1} and γ2\gamma_{2} at xx. Then

χ¯γ1,Tχ¯γ2,T=±χ¯α1,Tχ¯α2,Ti=1nyibT(T,τi)(ciai)/2±χ¯β1,Tχ¯β2,Ti=1nyibT(T,τi)(cibi)/2\overline{\chi}_{\gamma_{1},\mathcal{L}_{T}}\overline{\chi}_{\gamma_{2},\mathcal{L}_{T}}=\pm\overline{\chi}_{\alpha_{1},\mathcal{L}_{T}}\overline{\chi}_{\alpha_{2},\mathcal{L}_{T}}\prod_{i=1}^{n}y_{i}^{b_{T}(\mathcal{L}_{T},\tau_{i})(c_{i}-a_{i})/2}\pm\overline{\chi}_{\beta_{1},\mathcal{L}_{T}}\overline{\chi}_{\beta_{2},\mathcal{L}_{T}}\prod_{i=1}^{n}y_{i}^{b_{T}(\mathcal{L}_{T},\tau_{i})(c_{i}-b_{i})/2}

where ci=e(γ1,Li)+e(γ2,Li),ai=e(α1,Li)+e(α2,Li)c_{i}=e(\gamma_{1},L_{i})+e(\gamma_{2},L_{i}),a_{i}=e(\alpha_{1},L_{i})+e(\alpha_{2},L_{i}) and bi=e(β1,Li)+e(β2,Li)b_{i}=e(\beta_{1},L_{i})+e(\beta_{2},L_{i}).

Case Two: Intersection of Generalized Arc/Loop with Generalized Loop/One-sided Closed Curve

When γ1\gamma_{1} is a generalized arc or loop and γ2\gamma_{2} is a generalized loop, then Proposition 6.5 from [8] applies after lifting to the orientable double cover. When γ2\gamma_{2} is a one-sided closed curve, we note that the only new type of step in the standard MM-path is the elementary step of type 3’, from Definition 6.5, when going through the crosscap at the end, see Definition 6.7. Specifically, the steps of type 1 and 2 remain the same as in the orientable case, meaning we can directly apply the arguments from Lemma 6.10 and Proposition 6.5 from [8] to prove Proposition 7.2.

xxγ1\gamma_{1}γ2\gamma_{2}\Rightarrowα\alpha++β\betaxxγ1\gamma_{1}γ2\gamma_{2}\Rightarrowα\alpha++β\beta
Figure 19. Examples of intersections with curves and their resolutions.
Proposition 7.2.

Let γ1\gamma_{1} be a generalized arc or loop and let γ2\gamma_{2} be a generalized loop or one-sided closed curve, such that γ1\gamma_{1} and γ2\gamma_{2} intersect each other at least once; let xx be a point of intersection; and let α\alpha and β\beta be the two arcs/curves obtained by resolving the intersection of γ1\gamma_{1} and γ2\gamma_{2} at xx. Then

χ¯γ1,Tχ¯γ2,T=±χ¯α,Ti=1nyibT(T,τi)(ciai)/2±χ¯β,Ti=1nyibT(T,τi)(cibi)/2\overline{\chi}_{\gamma_{1},\mathcal{L}_{T}}\overline{\chi}_{\gamma_{2},\mathcal{L}_{T}}=\pm\overline{\chi}_{\alpha,\mathcal{L}_{T}}\prod_{i=1}^{n}y_{i}^{b_{T}(\mathcal{L}_{T},\tau_{i})(c_{i}-a_{i})/2}\pm\overline{\chi}_{\beta,\mathcal{L}_{T}}\prod_{i=1}^{n}y_{i}^{b_{T}(\mathcal{L}_{T},\tau_{i})(c_{i}-b_{i})/2}

where ci=e(γ1,Li)+e(γ2,Li),ai=e(α,Li)c_{i}=e(\gamma_{1},L_{i})+e(\gamma_{2},L_{i}),a_{i}=e(\alpha,L_{i}) and bi=e(β,Li)b_{i}=e(\beta,L_{i}).

Case Three: Non-simple Generalized Arc/Loop/One-sided Closed Curve

Just like the previous case, when γ\gamma is a generalized arc or closed curve with a self-intersection at xx, me may lift to the orientable surface and apply Proposition 6.6 from [8]. When γ\gamma is a one-sided closed curve, using the standard MM-path for one-sided closed curves, we may directly apply the arguments from Lemma 6.10 and Proposition 6.6 of [8] to conclude Proposition 7.3.

\Rightarrowα1\alpha_{1}α2\alpha_{2}γ\gammaxx++β\betaxx\Rightarrowγ\gammaα1\alpha_{1}α2\alpha_{2}++β\betaγ\gammaxx\Rightarrowα1\alpha_{1}α2\alpha_{2}++β\beta
Figure 20. Examples of self-intersection resolution
Proposition 7.3.

Let γ\gamma be a generalized arc, closed curve or one-sided closed curve with a self-intersection at xx. Let α1,α2\alpha_{1},\alpha_{2} and β\beta be the generalized arcs/loops/one-sided curves obtained by resolving the intersection at xx. Then

χ¯γ,T=±χ¯α1,Tχ¯α2,Ti=1nyibT(T,τi)(ciai)/2±χ¯β,Ti=1nyibT(T,τi)(cibi)/2\overline{\chi}_{\gamma,\mathcal{L}_{T}}=\pm\overline{\chi}_{\alpha_{1},\mathcal{L}_{T}}\overline{\chi}_{\alpha_{2},\mathcal{L}_{T}}\prod_{i=1}^{n}y_{i}^{b_{T}(\mathcal{L}_{T},\tau_{i})(c_{i}-a_{i})/2}\pm\overline{\chi}_{\beta,\mathcal{L}_{T}}\prod_{i=1}^{n}y_{i}^{b_{T}(\mathcal{L}_{T},\tau_{i})(c_{i}-b_{i})/2}

where ci=e(γ,Li),ai=e(α1,Li)+e(α2,Li)c_{i}=e(\gamma,L_{i}),a_{i}=e(\alpha_{1},L_{i})+e(\alpha_{2},L_{i}) and bi=e(β,Li)b_{i}=e(\beta,L_{i}).

Case Four: Intersection of Homotopic One-sided Closed Curves

The case that requires the most care occurs when we have two homotopic one-sided closed curves that intersect one another. It’s worth noting that any two curves homotopic to a one-sided closed curve will have at least one intersection point, meanwhile, in the orientable case, two homotopic two-sided curves can always be adjusted so that they are disjoint.

For the next proposition, we follow the notation from Proposition 4.6 of [1]. Namely, if α\alpha is a one-sided closed curve, then α2\alpha^{2} will denote the one-sided closed curve of multiplicity two, see Figure 21 below. In terms of the orientable double cover, one can interpret α2\alpha^{2} as the concatenation of the two lifts of α\alpha on the orientable surface, as such, α2\alpha^{2} is a two-sided closed curve enclosing the crosscap. The following proposition generalizes Proposition 4.6 of [1].

α\alphaα2\alpha^{2}
Figure 21. Depiction of α2\alpha^{2} on the Möbius band with two marked points
Proposition 7.4.

Let α\alpha be a one-sided closed curve, and let α2\alpha^{2} be the two sided-closed curve enclosing the crosscap obtained from resolving an intersection of α\alpha with another one-sided closed curve homotopic to α\alpha, then

(χ¯α,T)2=χ¯α2,T2(\overline{\chi}_{\alpha,_{T}})^{2}=\overline{\chi}_{\alpha^{2},T}-2
Proof.

Let α\alpha^{\prime} be the reflection of α\alpha about the crosscap, as in Lemma 6.11. We can write

χ¯α,T=tr([0xi11xi10]M¯),\overline{\chi}_{\alpha,T}=\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M}),

where M¯\overline{M} is the reduced standard MM-path of α\alpha with the final step excluded (so we do not go through the crosscap). Similarly, we can write

χ¯α,T=tr([0xi11xi10]M¯).\overline{\chi}_{\alpha^{\prime},T}=\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}}).

With this notation, we have χ¯α2,T=tr(M¯M¯)\overline{\chi}_{\alpha^{2},T}=\text{tr}(\overline{M^{\prime}}\cdot\overline{M}). With all of this information in mind, from earlier results from this paper, as well as other basic considerations, we have

  • χ¯α,T=χ¯α,T\displaystyle{\overline{\chi}_{\alpha,T}=\overline{\chi}_{\alpha^{\prime},T}}

  • det(M¯)=det(M¯)=1\displaystyle{\det(\overline{M})=\det(\overline{M^{\prime}})=1}

  • det([0xi11xi10])=1and[0xi11xi10]2=I2\displaystyle{\det(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix})=-1\quad\text{and}\quad\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}^{2}=I_{2}}

  • [0xi11xi10]M¯[0xi11xi10]=M¯\displaystyle{\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}}\cdot\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}=\overline{M}}

Lastly, when A,BA,B are 2×22\times 2 matrices with |det(A)|=|det(B)|=1|\det(A)|=|\det(B)|=1 and tr(A),tr(B)>0\text{tr}(A),\text{tr}(B)>0, we have

tr(A)tr(B)=tr(AB)+det(B)tr(AB1).\text{tr}(A)\text{tr}(B)=\text{tr}(AB)+\det(B)\text{tr}(AB^{-1}).

Putting everything together, we have

(χ¯α,T)2\displaystyle(\overline{\chi}_{\alpha,T})^{2} =χ¯α,Tχ¯α,T\displaystyle=\overline{\chi}_{\alpha,T}\cdot\overline{\chi}_{\alpha,T}
=χ¯α,Tχ¯α,T\displaystyle=\overline{\chi}_{\alpha^{\prime},T}\cdot\overline{\chi}_{\alpha,T}
=tr([0xi11xi10]M¯)tr([0xi11xi10]M¯)\displaystyle=\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}})\cdot\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M})
=tr([0xi11xi10]M¯)tr(M¯[0xi11xi10])\displaystyle=\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}})\cdot\text{tr}(\overline{M}\cdot\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix})
=tr([0xi11xi10]M¯M¯[0xi11xi10])tr([0xi11xi10]M¯(M¯[0xi11xi10])1)\displaystyle=\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}}\cdot\overline{M}\cdot\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix})-\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}}\cdot(\overline{M}\cdot\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix})^{-1})
=tr([0xi11xi10][0xi11xi10]M¯M¯)tr([0xi11xi10]M¯[0xi11xi10]M¯1)\displaystyle=\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}}\cdot\overline{M})-\text{tr}(\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M^{\prime}}\cdot\begin{bmatrix}0&&x_{i_{1}}\\ \frac{1}{x_{i_{1}}}&&0\end{bmatrix}\cdot\overline{M}^{-1})
=tr(M¯M¯)tr(M¯M¯1)\displaystyle=\text{tr}(\overline{M^{\prime}}\cdot\overline{M})-\text{tr}(\overline{M}\cdot\overline{M}^{-1})
=χ¯α2,Ttr(I2)\displaystyle=\overline{\chi}_{\alpha^{2},T}-\text{tr}(I_{2})
=χ¯α2,T2\displaystyle=\overline{\chi}_{\alpha^{2},T}-2

8. Appendix

In this appendix, we explicitly demonstrate the skein relation manipulations referred to in Section 2. We begin by demonstrating the missing step in the computation for the quasi-mutation depicted in the body of the paper. Namely, we show on the double cover why we can push a loop through the crosscap.

[Uncaptioned image]
[Uncaptioned image]

Included below is the skein relation derivation of the fourth quasi-mutation relation given in Definition 2.7. We restate the mutation relation for reference.

Given tTt\in T, the quasi-mutation of Σ\Sigma in the direction TT as the pair μt(T,𝐱)=(T,𝐱)\mu_{t}(T,\mathbf{x})=(T^{\prime},\mathbf{x^{\prime}}) where T=μt(T)=T{t}{t}T^{\prime}=\mu_{t}(T)=T\setminus\{t\}\sqcup\{t^{\prime}\} and 𝐱={xv|vT}\mathbf{x^{\prime}}=\{x_{v}\,|\,v\in T^{\prime}\} such that xtx_{t^{\prime}} is defined as follows: If tt is an arc separating a triangle with sides (a,b,t)(a,b,t) and an annuli with boundary tt and one-sided curve dd, then the relation is xtxt=(xa+xb)2+xd2xaxbx_{t}x_{t^{\prime}}=(x_{a}+x_{b})^{2}+x_{d}^{2}x_{a}x_{b}.

[Uncaptioned image]
[Uncaptioned image]

The above computation shows that this crossing resolution yields the relation xtxt=xa2+xb2+xbxaxcx_{t}x_{t}^{\prime}=x_{a}^{2}+x_{b}^{2}+x_{b}x_{a}x_{c} where cc is a two sided-closed curve enclosing the crosscap. It remains to show that xc=2+xd2x_{c}=2+x_{d}^{2}. Let ee be the arc passing through the crosscap. We now show the resolution of the crossing of the arc ee and the two-sided closed curve cc which encloses the crosscap, which yields the relation xcxe=2xe+xaxd+xbxdx_{c}x_{e}=2x_{e}+x_{a}x_{d}+x_{b}x_{d}.

[Uncaptioned image]

Finally, we show that xexd2=xaxd+xbxdx_{e}x_{d}^{2}=x_{a}x_{d}+x_{b}x_{d} by resolving the remaining crossings. With this computation, we have xcxe=2xe+xexd2x_{c}x_{e}=2x_{e}+x_{e}x_{d}^{2} implying that xc=2+xd2x_{c}=2+x_{d}^{2}. This completes the derivation of the fourth mutation relation from Definition 2.7.

[Uncaptioned image]

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