An algorithm to compute the fundamental classes of spin components of strata of differentials

Given a dual graph $\Gamma$ of some stable curve of genus $g$ and $n$ marked points, there is a natural clutching morphism

The strategy of our recursive algorithm is to compute $[\overline{\mathcal{H}}_{g}(\mu)]^{\operatorname{spin}}$ by solving a system of linear equations from clutching pullbacks, which reduce the problem to computing the spin stratum classes $[\overline{\mathcal{H}}_{g^{\prime}}(\mu^{\prime})]^{\operatorname{spin}}$ of smaller genera or fewer singularities. By proving that $\mathrm{H}^{2g-2}(\overline{\mathcal{M}}_{g,n};\mathbb{Q})=0$ in Theorem 4.1, we can strengthen a result of [AC98] (Proposition 5.1), which will guarantee the injectivity of the direct sum of clutching pullbacks with respect to one-edge stable graphs $\Gamma$

for $j$ within some appropriate range. By such a result, one can actually already construct a theoretical algorithm to compute the spin stratum classes. However, in practice, the cohomology of moduli spaces of stable curves $\overline{\mathcal{M}}_{g,n}$, for all $g,n$, is not entirely known. Thus, it is not possible to directly implement the strategy we mentioned above. It will only be feasible if we restrict our computation to the tautological ring $\operatorname{RH}^{*}(\overline{\mathcal{M}}_{g,n})$, on which we have a better understanding of the generators and relations. The drawback is that if a spin stratum class is not tautological, then our algorithm implemented in admcycles will raise errors. Thus, the first assumption we need to make for our implementation of the algorithm is that the spin stratum classes are tautological.

A bottleneck for programming the aforementioned strategy is that the clutching pullback with respect to the self-loop graph will not reduce the computation complexity (see Section 5.4). However, we observed that, in many cases where the number of marked points $n$ is larger equal to $3$, the injectivity of (1.1) holds when we restrict the clutching pullbacks to the tautological ring $\operatorname{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ even without including the non-compact type graph. This means that in the recursive computation of a spin stratum class, we only need to perform the clutching pullback with respect to the self-loop graph at most once. This ensures that the runtime of our algorithm will not be unreasonably long.

However, we are not able to prove that the phenomenon above is always true in this paper. Hence, we rephrase this into an assumption of the sufficiency of compact type clutching (Assumption 5.3), by which we mean that the direct sum of clutching pullbacks, restricting to $\operatorname{RH}^{*}(\overline{\mathcal{M}}_{g,n})$, with respect to one-edge graphs of compact type is injective for $g\geq 1,n\geq 3$ and for codimension $j$ which is not too large.

Our implemented algorithm for computing spin stratum classes in this paper will be based on two assumptions: First, the spin stratum classes are tautological; second, the assumption of the sufficiency of compact type clutching. For particular $g,n$ and $j$, it is known that the tautological classes generate the cohomology groups:

• •

  $g=0$ and for any $n$ and $j$ (cf. [Kee92]);

• •

  $g=1$ and $n<11$, for any $j$ (cf. [GP03]);

• •

  $g=2$ and $n<20$, and for any even number $j$ (cf. [Pet16]);

• •

  $g=3$ and $n=0,1,...,8$ (cf. [Ber08], [GL99], [Get98], [CL22]);

• •

  $g=4$ and $n=0,1,...,6$ (cf. [CL22]);

• •

  and more cf. [CL22].

Thus, in the cases listed above, the first assumption is automatically true. On the other hand, the second assumption has been checked by the Sage package admcycles on the tautological ring and it is true for $(g,n)$ equal to

To summarise, the assumptions we made are automatically true for the values of $g,n$ listed above. We have used the implemented algorithm to compute the spin stratum classes for plenty of signatures where $(g,n)$ is equal to

The results by the implemented algorithm for $(g,n)$ in the list above are guaranteed to be correct.

Lastly, we explain how we compute each individual clutching pullback. Indeed, we apply the Clutching Pullback Formula (5.3) of $[\overline{\mathcal{H}}_{g}(\mu)]^{\operatorname{spin}}$. The formula is derived by using the spin structure on the moduli space of multi-scale differentials $\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)$ (the definitions will be recalled in Section 2.2), which defines a compactification of the stratum of differentials $\mathbb{P}\Omega\mathcal{M}_{g,n}(\mu)$. The compactification $\mathbb{P}\Omega\mathcal{M}_{g,n}(\mu)\subset\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)$ has normal crossing boundary divisors and a proper morphism $p:\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)\longrightarrow\overline{\mathcal{M}}_{g,n}$ by sending a multi-scale differential to its underlying stable curve. The advantage to working with the compactification $\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)$ is that the set of connected components of $\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)$ is in bijection with the set of connected components of $\mathbb{P}\Omega\mathcal{M}_{g,n}(\mu)$, whereas in the Deligne-Mumford compactification the spin components can have intersections on the boundary. This implies that the closures of different spin components are still disjoint, thus the spin parity on a multi-scale differential of even type can be defined as the spin parity of the differentials lying in the neighbourhood. As a result, $\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)$ has spin components and

Note that deriving the Clutching Pullback Formula (5.3) requires an understanding of how the spin parity behaves on the boundary strata of $\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)$. The spin parity of a multi-scale differential lying on the boundary can be computed in the following way:

• •

  First, we undegenerate the multi-scale differential to a usual differential which can be realized by a flat surface

• •

  Then, we compute the spin parity by the explicit Formula (2.1) for the Arf invariant.

Our first observation on the spin parity of a multi-scale differential is the following proposition. We briefly define the objects involved in the following proposition (readers are referred to Section 2 for the details).

• •

  An enhanced level graph is a graph whose edges are weighted by positive integers (which we call enhancements) and there is a full order on the set of vertices (which can be realized by placing the vertices on different levels).

• •

  A twisted differential of type $\mu$ on a stable curve $(X;z_{1},...,z_{n})$ is a collection of non-zero meromorphic differentials on the irreducible components of $X$ such that their order of poles and zeros at $z_{1},...,z_{n}$ are prescribed by $\mu$. A twisted differential is compatible with an enhanced level graph $\Delta$ if $\Delta$ is the dual graph of $(X;z_{1},...,z_{n})$, and the orders of poles and zeros at the nodal points are prescribed by the enhancements (altered by $\pm 1$).

• •

  A (global) prong-matching is a collection of cyclic-order-reversing bijections which prescribe the matchings of horizontal geodesics (with respect to the meromorphic differentials) converging to the nodal points from two components.

The boundary divisors $D_{\Delta}$ in $\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu)$ are parametrized by enhanced level graphs $\Delta$ such that either the number of levels is $2$ and there is no horizontal edge (an edge connecting two vertices on the same level), or there is exactly one level and one horizontal edge. We denote the set of two-level graphs without horizontal edges of a given stratum by $\operatorname{LG}_{1}(\mu)$. For short, we call such a level graph $\Delta$ a vertical two-level graph. The degree of the map $p|_{D_{\Delta}}$ is equal to the number of prong matching equivalence classes $\frac{\prod_{e\in E(\Delta}\kappa_{e}}{\ell_{\Delta}}$, where $\ell_{\Delta}$ is the l.c.m. of the enhancements. In addition, the image $p(D_{\Delta})$ will be equal to the image of

where $\mathbb{P}\Xi\overline{\mathcal{M}}_{\Delta,\top}$ and $\mathbb{P}\Xi\overline{\mathcal{M}}_{\Delta,\bot}$ are the level strata corresponding to the top and bottom level of $\Delta$. We have the following corollary to Proposition 1.2. We define $[D_{\Delta}]^{\operatorname{spin}}=[D_{\Delta}^{+}]-[D_{\Delta}^{-}]\in\operatorname{CH}^{1}(\mathbb{P}\Xi\overline{\mathcal{M}}_{g,n}(\mu))$.

Although Corollary 1.3 will not be applied directly in the computation of the clutching pullback a spin stratum class, it will be used combining with the residue resolving (Proposition 5.9). Furthermore, the principle of Corollary 1.3 is used to derive the Clutching Pullback Formula of a spin stratum class. More precisely, if $\Delta$ has any edge of even enhancement, then the term corresponding to $\Delta$ in the Clutching Pullback Formula will be zero. Otherwise, the term corresponding to $\Delta$ will be just a product of the spin stratum classes on the top level and that on the bottom level.

Theorem 1.4 only guarantees a solution in $\mathrm{H}^{*}(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$. In practice, we work with the Sage package admcycles, by which we can only compute the solution in the tautological ring. Since admcycles uses the generalised Faber-Zagier relations (cf. [Pix12], [PPZ15],[Jan17]) to compute for a basis of the tautological ring. As a result, the correctness of our implemented algorithm will depend on the assumption that the generalised Faber-Zagier relations span the complete set of relations of the decorated strata classes (which serve as the additive generators) in the tautological ring. We have the following conditional result:

By the implemented algorithm, we have computed the spin stratum classes of plenty of different signatures for $(g,n)$ within the range (1.3), where the tautological ring is just the cohomology ring and the assumption of Faber-Zagier relations can be verified by Poincaré duality. Thus, in such range, our results are just the exact results.

First, we want to give an overview of the strategies to compute cycle classes $[\overline{\mathcal{H}}^{k}_{g}(\mu)]$ in $\operatorname{CH}^{*}(\overline{\mathcal{M}}_{g,n})$ or $\mathrm{H}^{*}(\overline{\mathcal{M}}_{g,n})$. In [Sau19], Sauvaget has constructed an algorithm to compute the stratum class of $1$-differentials $[\overline{\mathcal{H}}_{g}(\mu)]$ in $\mathrm{H}^{*}(\overline{\mathcal{M}}_{g,n})$ and he shows that the stratum classes are tautological. Later in [FP18], Farkas and Pandhahripande conjectured that the sum of the stratum class of $1$-differentials and the boundary terms is equal to the twisted double ramification cycle (the definition will be recalled in Section 2.2). In [Sch16], Schmitt extended this conjecture to $k$-differentials and gave a recursion to extract the stratum classes $[\overline{\mathcal{H}}^{k}_{g}(\mu)]$ from the conjectured equations. This conjecture has been finally proved by [HS21] and [BHPSS20].

In [CSS21], Constantini, Sauvaget and Schmitt have constructed the spin variant of the twisted double ramification cycles. In addition, they conjectured an equation that is similar to the one conjectured by Farkas and Pandharipande in [FP18], stating that the sum of the spin stratum class and some boundary terms is equal to the spin double ramification cycle. (The exact formulation will be recalled in Conjecture 7.2 in Section 7.3.) Within the range in List (1.2), the spin stratum classes we computed agree with Conjecture 7.2. Thus, the algorithm of our paper can help verify the conjecture in the range where computer power is sufficient to perform the computation.

Note that all the spin stratum classes of holomorphic strata of $1$-differentials are computed for $g\leq 4$, and they lie in the tautological ring (see Appendix B). Hence, Conjecture 7.2 will enable us to recursively compute the spin stratum classes $[\mathcal{H}^{k}_{g}(\mu)]^{\operatorname{spin}}$ by tautological classes. Hence, if we assume the conjecture about the spin stratum classes (Conjecture 7.2) to be true, then we have the following proposition:

In this subsection, we introduce the spin parity of a flat surface. First, we start with some terminology on a topological surface, then we will dive into the spin parity on a flat surface. By abusing symbols, we will denote a simple loop on a topological surface $X$ and its corresponding cycle in $\mathrm{H}_{1}(X,\mathbb{Z})$ both by the same letter.

The algebraic intersection number $i(\bullet,\bullet)$ induces a symplectic form on $\mathrm{H}_{1}(X,\mathbb{Z})$, where $X$ is a compact orientable surface. Two cycles $\alpha,\beta$ form a symplectic pair if their intersection number $i(\alpha,\beta)$ is $\pm 1$. Let the genus of $X$ be $g$. Then a generating set $\{\gamma_{1},...,\gamma_{2g}\}$ of $\mathrm{H}_{1}(X,\mathbb{Z})$ is a symplectic basis if it can be partitioned into symplectic pairs such that any two cycles from different symplectic pairs have zero intersection number.

Let $(X,\omega)$ be a flat surface induced by a differential $\omega$ on the compact Riemann surface $X$ and $\gamma:[0,1]\longrightarrow(X,\omega)$ be a smooth simple curve not passing through the conical singularities. On a flat surface, parallel translations along a smooth simple loop (avoiding the conical singularities) will preserve the direction of a vector. This implies that we can define the north direction at each point (except for the conical singularities) simultaneously and define a continuous map

There is a unique lift $\widetilde{T}_{\gamma}:[0,1]\longrightarrow\mathbb{R}$ if we fix the starting point on $\mathbb{R}$ to be the origin. We define the turning number or index $\operatorname{ind}(\gamma)$ to be $\widetilde{T}_{\gamma}(1)$. If $\omega$ is a differential of even type, then $\operatorname{ind}(\gamma)\pmod{2}$ will be invariant under the deformation of $\gamma$. Let $\{a_{1},...,a_{g},b_{1},...,b_{g}\}$ be a symplectic basis of $\mathrm{H}_{1}(X,\mathbb{Z})$. Then the parity of spin of a flat surface $(X,\omega)$ is defined as:

It has been shown in [Joh80] that the parity of spin does not depend on the choice of a symplectic basis and is a locally constant function on a stratum of differentials of even type. We have now defined the parity of spin on a compact flat surface. Actually, we can extend the definition to the case that the differential $\omega$ of the pair $(X,\omega)$ is meromorphic. Let $P$ be the set of poles of the meromorphic differential $\omega$, then one can realize $X\setminus P$ by a noncompact orientable surface carrying a flat metric with conical singularities. We call such a flat realization of a meromorphic differential a noncompact flat surface. A symplectic basis on $X$ can be realised by smooth simple closed loops on $X\setminus P$ which circumvent the conical singularities. By observation, one can see that if $\omega$ is of even type, the turning numbers $\pmod{2}$ are independent of the choice of realizations. Hence, the definition of spin parities in (2.1) can be also applied to noncompact flat surfaces associated to meromorphic differentials of even type (cf. [Boi15]).

We will in this subsection recall the definition of twisted $k$-differentials, enhanced level graphs and prong-matchings. In order to define $k$-twisted canonical divisors, we introduce twisted differentials and enhanced level graphs for general values of $k$. However, in the definition of multi-scale differentials and the rest of the text, we will mainly focus on the case that $k=1$.

In this text, a twisted differential simply refers to a twisted $1$-differential. Some crucial information of a twisted differential can be described by an enhanced level graph. An enhanced level graph $\Delta$ consists of:

• (i)

  a dual graph $\bar{\Delta}$ of $(X,\boldsymbol{z})$;

• (ii)

  a level function $\ell:V(\bar{\Delta})\longrightarrow\{0,-1,-2,...\}$ such that $\ell^{-1}(0)$ is nonempty;

• (iii)

  for each vertical edge $e$ an assigned positive integer $\kappa_{e}$ which is called enhancement.

We say that a twisted $k$-differential $\boldsymbol{\eta}$ is compatible with $\Delta$ if for the order on $V(\bar{\Delta})$ induced by $\ell$:

• (iv)

  Given an edge $e$ connecting vertices $v_{1},v_{2}$ in $\Delta$, $\ell(v_{1})=\ell(v_{2})$ if and only if $\operatorname{ord}_{q_{1}}(\eta_{v_{1}})=\operatorname{ord}_{q_{2}}(\eta_{v_{2}})=-k$. If $\ell(v_{1})>\ell(v_{2})$, then $\operatorname{ord}_{q_{1}}(\eta_{v_{1}})=\kappa_{e}-k$ and $\operatorname{ord}_{q_{2}}(\eta_{v_{2}})=-\kappa_{e}-k$.

If not specified, by saying that a twisted $1$-differential $\boldsymbol{\eta}$ is compatible with $\Delta$, the residues of the poles of $\boldsymbol{\eta}$ should also satisfy the following condition called the global residue condition (GRC). Since the GRC for general $k$ is technical and will not be used explicitly in our text, we will only state the GRC for $k=1$ here.

• (v)

  For every level $L$, a connected component $Y$, which has no leg representing a marked pole, of the subgraph $\Delta_{>L}$ (which is a graph consisting of edges and vertices above level $L$) will give us a residue condition. Let $e_{1},...,e_{m}$ be edges connecting $Y$ to vertices on level $L$. Then the residue condition induced by $Y$ is

  $$\sum_{j=1}^{m}\operatorname{res}_{q_{j}^{-}}(\eta_{e_{j}^{-}})=0,$$

  where $e_{j}^{-}$ is the lower vertex of edge $e_{j}$ and $q_{j}^{-}$ is the point in $X_{e_{j}^{-}}$ representing the node.

We will abuse the notation for edges so that we write $q$ for both a node in $X$ and also the corresponding edge in the compatible graph.

The rest of the text (except Section 7) will mainly focus on the case $k=1$, thus if not specified, a differential will refer to a $1$-differential.

Let $\boldsymbol{\eta}$ be a twisted differential. Each component $\eta_{v}$ of the twisted differential $\boldsymbol{\eta}$ endows $X_{v}$ with a flat structure. Moreover, for two points $q^{+},q^{-}$ in $\widetilde{X}=\bigsqcup_{v\in V(\Delta)}X_{v}$ corresponding to a vertical node $q$ (a node corresponding to a vertical edge), there will be the same number of horizontal geodesics converging to these points. We call these geodesics prongs. The horizontal geodesics (of direction to east) which run towards the vertical node are incoming prongs and the others are outgoing prongs. We denote the set of incoming prongs and the set of outgoing prongs at $x$ by $P^{in}_{x}$ and $P^{out}_{x}$ respectively. There is a natural cyclic order on the set of prongs. A local prong-matching at node $q$ is a cyclic-order-reversing bijection $\sigma_{q}:P^{in}_{q^{-}}\longrightarrow P^{out}_{q^{+}}$ and a global prong-matching is a collection $\boldsymbol{\sigma}$ of local prong-matchings at every vertical node.

Let $X$ be a stable curve; $\boldsymbol{\eta}$ be a twisted $1$-differential compatible with some enhanced level structure on the dual graph $\bar{\Delta}$ of $X$; $\boldsymbol{\sigma}$ be a global prong-matching. There is a natural action of the level rescaling group $\mathbb{C}^{L}$ on the pair $(\boldsymbol{\eta},\boldsymbol{\sigma})$ (here $L$ is the number of levels), as now we describe:

• (1)

  Let $\boldsymbol{d}=(d_{j})_{j=0,-1,...,-L+1}\in\mathbb{C}^{L}$. We define

  $\displaystyle\boldsymbol{d}\cdot\boldsymbol{\eta}=\big{(}\exp(i2\pi d_{\ell(v)})\cdot\eta_{v}\big{)}_{v\in V(\Delta)}.$

• (2)

  For each vertical node $q$, we let

  $\displaystyle\boldsymbol{d}\cdot\sigma_{q}:P^{in}_{q^{-}}\longrightarrow P^{out}_{q^{+}}$

  be the map $\sigma_{q}$ precomposed and postcomposed by the rotations by the angles $-2\pi\operatorname{Re}(d_{\ell(q^{-})})/\kappa_{q}$ and $2\pi\operatorname{Re}(d_{\ell(q^{+})})/\kappa_{q}$ respectively, so that $\boldsymbol{d}\cdot\sigma_{q}$ remains to be a prong-matching.

Let

be the natural inclusion. We call $\mathbb{C}^{L-1}$ the reduced level rescaling group. The action of $\mathbb{C}^{L-1}$ on $(\boldsymbol{\eta},\boldsymbol{\sigma})$ is induced by that of the level rescaling group.

We now introduce the generalised stratum. The generalised strata arise naturally if we consider each level of a (connected) enhanced level graph. It is a product of strata of differentials:

It parametrizes differentials on disconnected compact Riemann surfaces. Similar to a moduli space of multi-scale differentials which compactifies a stratum of differentials, there is a compactification of a (projectivized) generalised stratum $\mathbb{P}\Omega\mathcal{M}_{\boldsymbol{g},\boldsymbol{n}}(\boldsymbol{\mu})$, which we denote by $\mathbb{P}\Xi\overline{\mathcal{M}}_{\boldsymbol{g},\boldsymbol{n}}(\boldsymbol{\mu}):=\mathbb{P}\big{(}\prod_{\nu=1}^{k}\Xi\mathcal{M}_{g_{\nu},n_{\nu}}(\mu_{\nu})\big{)}$. According to [CMZ20a], $\mathbb{P}\Omega\mathcal{M}_{\boldsymbol{g},\boldsymbol{n}}(\boldsymbol{\mu})\subset\mathbb{P}\Xi\overline{\mathcal{M}}_{\boldsymbol{g},\boldsymbol{n}}(\boldsymbol{\mu})$ is a compactification with normal crossing boundary divisors. The boundary strata of $\mathbb{P}\Xi\overline{\mathcal{M}}_{\boldsymbol{g},\boldsymbol{n}}(\boldsymbol{\mu})$ are parametrized by (disconnected) enhanced level graphs.

Moreover, if we consider a level below zero, there are also extra residue conditions induced by the global residue condition. Hence, we also introduce the notion of a generalised stratum with residue conditions $\Omega\mathcal{M}^{\mathfrak{R}}_{\boldsymbol{g},\boldsymbol{n}}(\boldsymbol{\mu})$. Here, $\mathfrak{R}$ refers to the residue space cut out by a set of residue conditions. Let $H_{p}$ be the set of pairs $(\nu,i)$ where $i$ is a marking of the component $\nu$ such that the marking $i$ is a pole. A collection $\mathfrak{R}$ of residue conditions consists of a partition $\boldsymbol{\lambda}_{\mathfrak{R}}$ of $H_{p}$ with parts $\lambda^{(j)}$. A part $\lambda^{(j)}$ represents a residue equation

Thus,

In this subsection, we will briefly introduce the welded surface and plumbed surface associated to a multi-scale differential. We refer the readers to [BCGGM19] for more details. Then we will define the spin parities on them and show why these two spin parities coincide with each other.

Given a multi-scale differential $(\boldsymbol{\eta},\Delta,\boldsymbol{\sigma})$, a welded surface is a topological surface $\overline{X}_{\boldsymbol{\sigma}}$ with a map $f:\overline{X}_{\boldsymbol{\sigma}}\longrightarrow X$ such that

• •

  the preimage of a node is a simple loop which we call a seam;

• •

  away from the seams, $f$ is an isomorphism;

• •

  the glueing of two boundaries circles of the preimages of two irreducible components of $X$ (to form a seam) is determined by the prong-matchings.

For instance, the compact closed surface in Figure 2 is a welded surface associated to the nodal curve in Figure 1. Notice that the twisted differential $\boldsymbol{\eta}$ endows $\overline{X}_{\boldsymbol{\sigma}}$ with a piecewise flat structure. From now on, if not specified, the welded surface associated to a multi-scale differential will refer to the topological welded surface plus the piecewise flat structure on it.

Now we will briefly introduce the plumbing surfaces. According to [BCGGM18], given a multi-scale differential $(\boldsymbol{\eta},\Delta,\boldsymbol{\sigma})$ such that $X$ is not smooth, one can construct a family of flat surfaces which is controlled by a collection of parameters $t_{-1},...,t_{-L+1}\in\mathbb{C}$ indexed by the levels below zero. A generic fiber is a connected flat surface, while the fiber over $t_{-1}=...=t_{-L+1}=0$ is the disconnected flat surface associated to the twisted differential $\boldsymbol{\eta}$. This deformation family of flat surfaces is called a plumbing family. A connected fiber of the deformation family will be called a plumbed surface. A plumbed surface is just a flat surface and its spin parity is defined as that of a flat surface. The construction of a plumbed surface is actually quite technical, nevertheless, if we only want to make arguments on the spin parity, it suffices to just consider the welded surface.

A welded surface $\overline{X}_{\boldsymbol{\sigma}}$ is an orientable closed surface. By considering the piecewise flat structure on the welded surface, one can still calculate the turning number of a smooth simple loop if the curve is transverse to the seams.

By the above proposition, we can determine the parity of a multi-scale differential of even type by computing (2.1) on the welded surface with respect to simple closed curves transverse to seams.

Let $\overline{X}_{\boldsymbol{\sigma}}$ be a welded surface compatible with a level graph $\Delta$. A cycle $\gamma$ on $\overline{X}_{\boldsymbol{\sigma}}$ is called a graph cycle if it can be realized as a simple loop $\gamma$ such that the geometric intersection numbers $\langle\gamma,\delta\rangle\leq 1$ for each seam $\delta$ but not all the intersection numbers are zero. A cycle $\delta$ on $\overline{X}_{\boldsymbol{\sigma}}$ is called a vanishing cycle if it is the homology class of some seam. A cycle which can be realised by a simple loop not intersecting any seam is called a non-crossing cycle.

Note that a graph cycle $\gamma$ induces a loop on $\Delta$ which consists of edges whose corresponding seam intersects with $\gamma$.

This subsection is to prove the existence of a $\Delta$-adapted symplectic basis. Notice that we are arguing just by topological properties of a nodal curve and its dual graph. Thus, for every topological welded surface, our claim of the existence of a $\Delta$-adapted symplectic basis still holds. To construct a $\Delta$-adapted symplectic basis on the welded surface $\overline{X}_{\boldsymbol{\sigma}}$, it is equivalent to finding a basis of $\mathrm{H}_{1}(\Delta)$. However, not every spanning tree can give us a set of fundamental cycles that can be completed to a $\Delta$-adapted symplectic basis. Namely, the simple closed curves on the welded surface, which realize the fundamental cycles of the graph, cannot be chosen to be disjoint. The following example will illustrate the problem.

We now show how this problem can be circumvented. Suppose we are given a spanning tree $T_{0}$ of a graph $\Delta$, we can apply the following iteration to obtain a tree $T$. This tree will induce a collection of graph cycles such that on each $C_{v}$ the nodal marked points and the segments connecting them form a planar graph. The main idea is to reduce the number of edges in a spanning tree adjacent to a vertex.

1. (1)

   Take an initial vertex $v^{*}$ such that it is only connected to one edge in $T_{0}$. The vertex $v^{*}$ is fixed throughout the iteration. On $T_{k}$, for every vertex $v$ except $v^{*}$, there is exactly one edge adjacent to $v$ going towards $v^{*}$, which we denote as $e^{(k)}_{v}$ and others are all going away from $v^{*}$.

2. (2)

   Let $\{q^{(k)}_{1},...,q^{(k)}_{g^{\prime}}\}$ be edges of $\Delta$ not included in $T_{k}$ and $c_{q^{(k)}_{i}}$ the fundamental cycles formed by adding $q^{(k)}_{i}$ in the tree. Let $w$ be a vertex of distance $k+1$ to $v^{*}$ in $T_{k}$. For any two edges adjacent to $w$, which are not $e^{(k)}_{w}$ and share a fundamental cycle $c_{q^{(k)}_{i}}$, we replace one of these two edges by $q^{(k)}_{i}$. The edge $q^{(k)}_{i}$ should be connected only to vertices that are of distance greater equal $k+1$ to $v^{*}$. We carry out this operation until all the vertices of distance $k+1$ to $v^{*}$ in $T_{k}$ have no going-away edges sharing a fundamental cycle. The resulting tree will be set to be $T_{k+1}$.

3. (3)

   Notice that the set of vertices of distance $k^{\prime}$ to $v^{*}$ in $T_{k}$, where $k^{\prime}\leq k$, will be also the set of vertices of distance $k^{\prime}$ to $v^{*}$ in $T_{k+1}$. Thus our operation in (2) will not change the distances we fixed in earlier stages. We terminate the iteration when we have exhausted all the vertices.

Let $\Sigma_{v}$ be the graph whose vertices are the nodal marked points on $C_{v}$, while its edges are segments connecting them. Now we show that the final spanning tree $T$ from the iteration above will give us on each $C_{v}$ a graph $\Sigma_{v}$ that is planar. First, notice that there are 3 types of vertices in $\Sigma_{v}$ which represent:

• (i)

  a node which is represented by an edge adjacent to vertex $v$ in the tree $T$ towards the initial vertex $v^{*}$,

• (ii)

  a node which is represented by an edge adjacent to vertex $v$ in the tree $T$ away $v^{*}$ and

• (iii)

  a node which is represented by an edge, not in the tree $T$

If the $\Sigma_{v}$ is induced by the result spanning tree $T$ from the iteration, then the vertices of the same type are not connected. There is only one vertex of type (i) in $\Sigma_{v}$ and every vertex of type (iii) in $\Sigma_{v}$ has only one edge to some vertex of type (ii). Moreover, the unique vertex of type (i) is only connected to vertices of type (ii). Thus, any cycle in $\Sigma_{v}$ can only consist of multiple edges between the unique vertex of type (i) and a vertex of type (ii). This kind of $\Sigma_{v}$ is always planar. Indeed, if we regard the multiple edges between the unique vertex of type (i) and a vertex of type (ii) as one edge, the resulting graph $\Sigma_{v}^{\prime}$ will be a tree. As adding an extra edge between two vertices will also add an extra face, the Euler characteristic will not change. Hence, the graph $\Sigma_{v}$ induced by the resulting tree $T$ is planar. Hence, we can now assert the existence of a $\Delta$-adapted symplectic basis on a welded surface. Finally, we can present our proof of Proposition 3.5.

In this subsection, we will describe how the spin parity changes when we change the prong-matching at a node.

The finite group $P_{\Gamma}=\prod_{e\in E(\Delta)}\mathbb{Z}/\kappa_{e}\mathbb{Z}$ is called the prong rotation group. Now let $r_{q}=(0,..,0,l,0,...0)$ (only the component for $e=q$ is $l$), where $\kappa_{q}>l>0$, be a representative of an element in the prong rotation group. Then it acts on a global prong-matching $\boldsymbol{\sigma}$ by altering the local prong-matching $\sigma_{q}$. It precomposes the cyclic-order-reversing bijection $\sigma_{q}:P^{in}_{q^{-}}\longrightarrow P^{out}_{q^{+}}$ with a rotation by $\frac{2\pi l}{\kappa_{q}}$ (in the ascending direction with respect to the cyclic order on $P^{in}_{q^{-}}$ ).

We denote the new welded surface obtained by applying $r_{q}$ on the prong-matchings be $\overline{X}_{r_{q}\boldsymbol{\sigma}}$. Let $\gamma$ on $\overline{X}_{\boldsymbol{\sigma}}$ be a loop representing a graph cycle passing through the seam corresponding to $q$. After we apply the prong rotation $r_{q}$, $\gamma$ will be a broken loop on $\overline{X}_{r_{q}\boldsymbol{\sigma}}$. By connecting the two ends of the broken loop by an arc of the seam cycle of $q$ we will get back a loop $r_{q}\gamma$ on $\overline{X}_{r_{q}\boldsymbol{\sigma}}$. By smoothing the loop, one can easily see that $\operatorname{ind}(\gamma)-\operatorname{ind}(r_{q}\gamma)=l\pmod{2}$.