Cluster algebras and skein algebras for surfaces

Let $\Sigma$ be an oriented surface with a finite set of marked points, which we distinguish as boundary marked points or interior punctures. The main goals of this paper are to review and clarify relationships between several generalizations of the cluster algebra and the skein algebra of $\Sigma$, and to explore the structural implications of this relationship.

The cluster algebra $\mathcal{A}(\Sigma)$ of a surface are a large class of cluster algebras that are particularly well-studied (See [Wil14] for an excellent survey). Cluster algebras are commutative algebras whose generators (cluster variables) and relations (cluster seeds) satisfy mutation properties determined by certain types of combinatorial data [FZ02]. In the case of the cluster algebra of a surface with boundary marked points, the generators correspond to edges of an ideal triangulation, and mutation encodes the combinatorial changes seen when a diagonal edge of the triangulation is flipped [GSV05, FG06]. The definition was motivated by the similarity between the mutation formulas for cluster algebras and the Ptolemy relation for lengths of edges seen in Teichmüller theory, especially as in the work of Penner [Pen87, Pen12]. In the presence of interior marked points, one can also define the cluster algebra, but one needs to combinatorially extend the definition of a triangulation and the edge flip move [FST08, FT18].

The cluster algebra for a surface has been found to be closely related to the Kauffman bracket skein algebra $\mathcal{S}_{q}(\Sigma)$ from quantum topology. The skein algebra was introduced in [Prz91, Tur91] as a generalization of the Jones polynomial [Jon85, Kau87] and appears prominently in the associated topological quantum field theory [BHMV95]. The skein algebra is generated by framed loops in $\Sigma\times(-1,1)$ modulo the Kauffman bracket skein relations, and it is non-commutative except for a few small surfaces and except when $q=\pm 1$. The skein algebra holds a special place in quantum topology due to its connections to hyperbolic geometry and algebraic geometry. In particular, it is a deformation quantization of the $\mathrm{SL}_{2}(\mathbb{C})$-character variety of the surface [Bul97, PS00, Tur91], which contains the Teichmüller space of $\Sigma$.

Because of the common connection to Teichmüller theory, it is natural to expect that there is some compatibility between the cluster algebra $\mathcal{A}(\Sigma)$ and the skein algebra $\mathcal{S}_{q}(\Sigma)$. Actually, because the cluster algebra is generated by edges of an ideal triangulation, a little thought shows that the compatibility should exist with versions of the skein algebra that include arcs between marked points on $\Sigma$. One such generalization appeared in work of Roger and Yang [RY14], who wanted to identify their skein algebra as a deformation quantization of Penner’s decorated Teichmüller space for a punctured surface $\Sigma$ [Pen87, Pen12]. At around the same time, Muller [Mul16] defined a different skein algebra, coming from a quantum cluster algebra for a surface of boundary marked points. Since then, these definitions have been combined and extended to include decorations of $\pm$ (called states) on the boundary marked points [Lê18, BKL24]. We aim to make a precise statement about their relationship with each other and with cluster algebras here.

We remark that similar compatibility results have been proved previously, with different levels of generality and different methods [Mul16, MQ23, MW24]. However, clear statements about the compatibility of the cluster algebra and skein algebra have been missing so far. The known compatibility results have treated only special cases (such as surfaces with only boundary marked points, or only with interior punctures) and only with certain variations. A goal of this paper is to provide general statements that incorporate what is known so far, both by providing references and by explaining how to extend existing proofs.

We begin with definitions of the various generalizations of the skein algebra that include arcs in Section 2. Our primary interest is in the Muller-Roger-Yang generalized skein algebra $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ from [BKL24]. We also discuss the Lê-Roger-Yang generalized skein algebra $\mathcal{S}_{q}^{\mathrm{LRY}}(\Sigma)$, which includes states at the boundary marked points. In Section 4, we prove a striking common property, that the skein algebras are finitely generated. In contrast, as we discuss in Section 6.2, the cluster algebra $\mathcal{A}(\Sigma)$ is not always finitely generated.

It was shown that the original skein algebra $\mathcal{S}_{q}(\Sigma)$ was finitely generated in [Bul99], and the proof was extended in for specific cases [Mul16, BPKW16, MW24]. Our proof here applies to the most general case. It immediately follows that some variations of generalized skein algebras are also finitely generated (Corollary 4.2 and 4.5).

In Section 5, we define the cluster algebra for surfaces with both boundary marked points and interior punctures, as well its associated upper cluster algebra $\mathcal{U}(\Sigma)$. We prove the compatibility of the cluster algebras and skein algebras in Section 6. To do so, we make two modifications. First, observe that the set of boundary edges form a multiplicative subset, and so we may define a localization $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)[\partial^{-1}]$. Then we set $q=1$ to obtain a commutative skein algebra $\mathcal{S}_{1}^{\mathrm{MRY}}(\Sigma)[\partial^{-1}]$, and we show that it contains a copy of the cluster algebra. Secondly, we define a certain subalgebra $\mathcal{S}_{q}^{\square}(\Sigma)$ of $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)[\partial^{-1}]$ that is generated by tagged arcs and loops; see Section 6 for the definition. We show that it is contained inside the upper cluster algebra $\mathcal{U}(\Sigma)$.

We remark that the second part of Theorem B also appears in [MQ23, Proposition 3.7]. However, at least to the authors, the extension from an unpunctured surface case [Mul16] to punctured surfaces needs some verification. This is because there are no tagged arcs in $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$, but there are vertex classes. We give some brief outline of the argument and references to relevant literature in Section 6.

Due to limited scope and time, the authors decided to restrict discussion here to the generalized skein algebras $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ and $\mathcal{S}_{q}^{\mathrm{LRY}}(\Sigma)$, and their variations. The research on the ordinary skein algebra $\mathcal{S}_{q}(\Sigma)$ is both long and rich. The results for the $\mathcal{S}_{q}(\Sigma)$ obviously inspired the theorems presented in this paper, as well as the open questions from Section 7. Unfortunately, a full discussion and bibliography are outside the scope of our paper. We encourage interested readers to check the references, e.g. [LY21, PBIMW24].

Let $\underline{\Sigma}$ be a compact oriented surface with possibly nonempty boundary $\partial\underline{\Sigma}$, and $V$ be a finite set of points of $\underline{\Sigma}$. The pair $\Sigma:=(\underline{\Sigma},V)$ is called a marked surface. From now on, if there is no chance of confusion, a surface is always a marked surface. We set $V_{\partial}:=V\cap\partial\underline{\Sigma}$ and call it by the set of boundary marked points. The complement $V_{\circ}:=V\setminus V_{\partial}$ is called the set of interior punctures. Depending on how to treat the arc classes meeting $\partial\underline{\Sigma}$, we use two different types of tangles.

If $\alpha$ is a manifold without boundary, both a $V$-tangle and a $\partial$-tangle are disjoint unions of framed loops on $\mathrm{Int}\;\Sigma\setminus V$.

Here, we define two generalized skein algebras. For each interior puncture $v_{i}\in V_{\circ}$, we assign a formal variable $v_{i}$. We fix the Laurent polynomial ring $\mathbb{C}[v_{i}^{\pm 1}]$ as the coefficient ring. It commutes with all other elements that we will introduce from now on. We also fix $q\in\mathbb{C}^{*}$.

The second generalized skein algebra is based on $\partial$-tangles and extra states. A state of a $\partial$-tangle $\alpha$ is a map $s:\partial\alpha\cap\partial\underline{\Sigma}\to\{+,-\}$. A stated $\partial$-tangle is a pair $(\alpha,s)$. If there is no confusion, we denote it by $\alpha$.

We will also consider several variations of $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ and $\mathcal{S}_{q}^{\mathrm{LRY}}(\Sigma)$ that have appeared in the literature. First, there are the subalgebras generated by tangles that avoid interior punctures. Let $\mathcal{S}_{q}^{\mathrm{M}}(\Sigma)$ be the subalgebra of $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ generated by all $V$-tangles that do not intersect any interior punctures. Similarly, $\mathcal{S}_{q}^{\mathrm{L}}(\Sigma)$ is the subalgebra of $\mathcal{S}_{q}^{\mathrm{LRY}}(\Sigma)$ generated by all $\partial$-tangles not meeting interior punctures. Note that relation (C) holds in $\mathcal{S}_{q}^{\mathrm{M}}(\Sigma)$ and $\mathcal{S}_{q}^{\mathrm{L}}(\Sigma)$.

The next variations drop relation (C). Let $\mathcal{S}_{q}^{\mathrm{M}+}(\Sigma)$ (resp. $\mathcal{S}_{q}^{\mathrm{L}+}(\Sigma)$) be a $\mathbb{C}$-algebra generated by all $V$-tangles ($\partial$-tangles) that do not meet any interior punctures modulo the relataions (A), (B), (D), (E), and (F) (resp. (A), (B), (D), (E’), and (F’)). Then we have morphisms

and

The first map $\pi$ is an epimorphism and $i$ is a monomorphism.

The last variation we consider involves a quotient of the stated skein algebras. For each boundary marked point $p$, a $\partial$-tangle in Figure 1 around $p$ is called a bad arc.

We may define similar quotients $\overline{\mathcal{S}}_{q}^{\mathrm{L}}(\Sigma)$ (resp. $\overline{\mathcal{S}}_{q}^{\mathrm{L}+}(\Sigma)$) of $\mathcal{S}_{q}^{\mathrm{L}}(\Sigma)$ (resp. $\mathcal{S}_{q}^{\mathrm{L}+}(\Sigma)$) by the bad arc classes. We will retain the same notation $p$ for the quotient map.

In Section 2, we introduced many variations of the skein algebra. From the algebraic perspective, they are not altogether ‘different’ skein algebras, and in this section, we will give a precise relationship between them. Our goal will be to justify the following commutative diagram.

From their definitions, we already have the maps $\pi$, $p$, and $i$ in the diagram. The morphisms $\pi$ and $p$ are epimorphisms, and $i$ is a monomorphism. We now define the morphisms $m$ and $r$.

The homomorphism

comes from the ‘moving trick.’ Recall that $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ is generated by $V$ -tangles, which are one-dimensional submanifolds embedded in $\Sigma\times(-1,1)$. So for each $v\in V$, the heights at the ends are all distinct. For each $v\in V_{\partial}$, we may draw a local diagram around $v$ so that all arcs ending at $v$ are not tangent to each other, as the diagram on the left in Figure 2. Then, by sliding the ends to the right component of $v$, we obtain a $\partial$-tangle. For each end $\alpha\cap\partial\underline{\Sigma}\times(-1,1)$, we assign a positive state. Using the skein relations in Definition 2.2, one may check that this is a well-defined map and indeed an algebra homomorphism.

The map $m$ is an identity around the interior punctures. Thus, by the same argument, we have similar homomorphisms

We will use the same letter $m$ to denote the moving trick homomorphism, to avoid introducing too many notations. Because of the existence of negative states, unless $\partial\underline{\Sigma}=\emptyset$, $m$ is not surjective.

We next define the map $r$. Let $\partial$ be the multiplicative subset of $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ generated by boundary arc classes, which are connected $V$-tangles (hence a single arc) joining two boundary marked points on the same connected component $D$ of $\partial\Sigma$ and isotopic to the boundary edge connecting two adjacent marked points on $D$. Since any boundary arc class is $q$-commutative with any $V$-tangles, $\partial$ satisfies the Ore condition [MR01, Section 2.1], hence we may define the Ore localization $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)[\partial^{-1}]$.

The proof of Proposition 3.1 does not affect the area near the interior punctures. Thus, the same proof works for $\mathcal{S}_{q}^{\mathrm{M}+}(\Sigma)$ and $\mathcal{S}_{q}^{\mathrm{M}}(\Sigma)$ as well. In summary, we have the equalities in Equation (3).

In this section, we prove that the skein algebras we consider are always finitely generated. In a way, this is rather surprising— as we will see later, the skein algebra contains a copy of a cluster algebra, but in many cases the cluster algebra is not finitely generated. Here we give a skein theoretic proof, extending [Bul99] and [BKWP16], where the finite generation of $\mathcal{S}_{q}(\Sigma)$ and $\mathcal{S}_{q}^{\mathrm{RY}}(\Sigma)$ were shown. Alternatively, one may prove the finite generation via quantum trace, as done in [BKL24] for $\overline{\mathcal{S}}_{q}^{\mathrm{LRY}}$.

Our strategy is based on the following observation. For a marked surface $\Sigma$, the interior punctures in $V_{\circ}$ can be drawn on the boundary of a small circular disk $D$ on $\mathrm{int}\;\Sigma$ (Figure 3). Then we may understand $\Sigma$ as a union of a small open neighborhood $\widetilde{D}$ of $D$ and the outside of $D$, that is, $\Sigma\setminus\mathrm{int}\;D$. The latter is a surface without interior puncture – now all marked points are on the boundary. We show that $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ is generated by the arc classes in $D$ and the curve classes whose supports are in $\Sigma\setminus\mathrm{int}\;D$. The former has an explicit finite set of generators [ACDHM21, Section 5]. For the latter, we may employ an extension of the algorithm in [Bul99].

Our discussion will be focused on simple curves, which are curves without any intersections in its diagram. Skein relations (A) and (D) imply that the $\mathbb{C}[v_{i}^{\pm 1}]$-algebra $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ is generated by simple curves.

For a simple curve $\alpha$, let $c(\alpha)$ be the number of connected components of $\mathrm{int}\;D\setminus\alpha$. We may extend the definition to curves in the isotopy class of $\alpha$, by taking $c(\alpha)$ as the minimum of the possible numbers of connected components. The geometric intersection number of $\alpha$ and $D$ is $i(\alpha):=c(\alpha)-1$, and $i(\alpha)=0$ if $\alpha$ does not meet $\mathrm{int}\;D$.

We may assume that $V_{\circ}\subset\partial D$ are arranged counterclockwise with respect to the center of $D$. Let $\beta_{ij}$ be the straight line segment connecting $v_{i}$ and $v_{j}$.

Now, it is sufficient to show that any skeins whose supports are in $\Sigma\setminus\mathrm{int}\;D$ is finitely generated. For this purpose, by stretching the boundary of $D$, we may draw $\Sigma\setminus\mathrm{int}\;D$ as in the right figure in Figure 6. This diagram is obtained by attaching several handles on a rectangular strip, and all marked points are on the strip itself, not on attached handles. In particular, there is no marked points on the handles. The long dashed boundary is $\partial D$, and the original boundary of $\Sigma$ is drawn on the top row.

Suppose that $\alpha\in\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ is a simple connected curve whose support is in $\Sigma\setminus\mathrm{int}\;D$. We fix an arbitrary orientation on $\alpha$. Assume that there is a handle $H$ on $\Sigma$ such that $\alpha$ traverses $H$ multiple times in the same direction. See the left figure in Figure 7. Then an ordinary skein relation can be applied to rewrite the curve in terms of curves that traverse $H$ strictly fewer times (Figure 7). Note that the rightmost diagram is a product of two disjoint curve classes, and each of these curves traverses $H$ strictly fewer times than the original diagram on the left hand side of the equation. By applying this argument in pairs, we may assume that $\alpha$ travels each handle at most twice, and if so, it travels exactly twice in the opposite direction.

Next, suppose that there is a handle $H$ where $\alpha$ travels twice, in opposite directions (top left of Figure 8). We may employ the ordinary skein relations three times, to write it as a combination of curves which traverse the handle at most once. See Figure 8.

Note that Bullock applied the above algorithm only to loops in [Bul99]. However, the argument works equally well for arc classes. This is because the resolutions in Figures 7 and 8 do not change the part of the arc outside $H$ (on the parts of the arc diagram that are dotted in the figures).

In summary, any simple curve can be generated by simple curves that traverse each handle at most once. These curves can be described by choosing the finite sequence of traveling handles in the case of loops and by choosing the initial/terminal marked points and the sequence of traveling handles in the case of arcs. (We do not claim that all sequences of handles will give simple curves and arcs – they may induce self-intersections.) Thus, there are only finitely many possibilities. As a result, we obtain the proof of Theorem A.

If we remove each interior puncture by a new boundary component without a marked point, the same proof of Theorem A works for $\mathcal{S}_{q}^{\mathrm{M}+}(\Sigma)$. Then we also obtain that $\mathcal{S}_{q}^{\mathrm{L}+}(\Sigma)$ is finitely generated in the same way. Any homomorphic image of a finitely generated algebra is also finitely generated, so we obtain the following corollary.

We next consider the cluster algebra $\mathcal{A}(\Sigma)$ coming from the curves in a triangulable oriented surface $\Sigma$. In this section, we give a brief definition of $\mathcal{A}(\Sigma)$. A more detailed description and further properties of $\mathcal{A}(\Sigma)$ can be found in [FST08, FT18]. In Section 6, we will explain its relationship with the skein algebra.

We now prove Theorem B, starting with a review of relevant definitions.