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Skein relations on punctured surfaces

Michael Tsironis

TL;DR

This work develops skein relations for cluster algebras arising from punctured surfaces, extending the classic surface cluster framework to interiors punctures. A key contribution is a combinatorial-algebraic bridge that links loop graphs to loop modules and loop strings, enabling explicit bijections between graphical and module-theoretic data. Using this framework, the author proves case-by-case skein relations for all configurations of crossings and taggings, and shows how products of cluster variables decompose into smoothed configurations with appropriate coefficient monomials. The results yield bases with positivity/compatibility properties for punctured-surface cluster algebras and illuminate the structural role of loop-graph/loop-module correspondences in the cluster-algebra landscape. Altogether, the work broadens the class of surface-cluster algebras for which skein relations and positivity-based bases can be constructed, by incorporating interior punctures and tagged-arc phenomena.

Abstract

This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce skein-type identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corresponding to compatible arcs. Incompatibility arises from phenomena such as intersections, self-intersections, and opposite taggings at punctures. To establish these identities, we develop a combinatorial-algebraic framework that relates loop graphs to certain representations. These skein relations can then be applied to investigate structural properties of cluster algebras from punctured surfaces. In particular, they can be used to prove the existence of bases satisfying natural positivity and compatibility conditions. This extends existing work on surface cluster algebras by incorporating punctures in the interior of the surface, thereby enlarging the class of cluster algebras for which such skein relations and bases can be constructed.

Skein relations on punctured surfaces

TL;DR

This work develops skein relations for cluster algebras arising from punctured surfaces, extending the classic surface cluster framework to interiors punctures. A key contribution is a combinatorial-algebraic bridge that links loop graphs to loop modules and loop strings, enabling explicit bijections between graphical and module-theoretic data. Using this framework, the author proves case-by-case skein relations for all configurations of crossings and taggings, and shows how products of cluster variables decompose into smoothed configurations with appropriate coefficient monomials. The results yield bases with positivity/compatibility properties for punctured-surface cluster algebras and illuminate the structural role of loop-graph/loop-module correspondences in the cluster-algebra landscape. Altogether, the work broadens the class of surface-cluster algebras for which skein relations and positivity-based bases can be constructed, by incorporating interior punctures and tagged-arc phenomena.

Abstract

This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce skein-type identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corresponding to compatible arcs. Incompatibility arises from phenomena such as intersections, self-intersections, and opposite taggings at punctures. To establish these identities, we develop a combinatorial-algebraic framework that relates loop graphs to certain representations. These skein relations can then be applied to investigate structural properties of cluster algebras from punctured surfaces. In particular, they can be used to prove the existence of bases satisfying natural positivity and compatibility conditions. This extends existing work on surface cluster algebras by incorporating punctures in the interior of the surface, thereby enlarging the class of cluster algebras for which such skein relations and bases can be constructed.
Paper Structure (20 sections, 18 theorems, 40 equations, 13 figures)

This paper contains 20 sections, 18 theorems, 40 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cluster Algebras
  4. Punctured surfaces
  5. Loop graphs
  6. Expansion formula of tagged arcs
  7. Quivers associated to loop graphs
  8. Abstract strings and loopstrings
  9. Construction of the loop graph of an abstract loopstring
  10. Construction of the loopstring of a loop graph
  11. Quiver representations and loop modules
  12. String modules
  13. Loop modules
  14. Bijection of loop graphs and loopstrings
  15. Minimal and maximal perfect matchings
  16. ...and 5 more sections

Key Result

Theorem 1.0.1

Let $A=kQ^\bowtie/I$ and $M(w)$ be a loop module over $A$ with associated loop graph $\mathcal{G}^\bowtie$. Then $\mathcal{L}(\mathcal{G})$, which denotes the perfect matching lattice of $\mathcal{G}$ is in bijection with the canonical submodule lattice $\mathcal{L}(M(w))$.

Figures (13)

  • Figure 1: The exchange graph of the cluster algebra of type $A_2$.
  • Figure 2: Self folded triangle $rlr$ and the associated tagged arc $r^\bowtie$, where $x_l= x_r x_{r^\bowtie}$ in the cluster algebra.
  • Figure 3: Flipping of the diagonal of a quadrilateral.
  • Figure 4: Flipping of a radius of a self-folded triangle.
  • Figure 5: Sign function on a tile of a snake graph where $\epsilon= \pm$.
  • ...and 8 more figures

Theorems & Definitions (84)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Definition 2.1.1
  • Definition 2.1.2
  • Example 2.1.3
  • Definition 2.2.1
  • Remark 2.2.2
  • Definition 2.2.3
  • Remark 2.2.4
  • Remark 2.2.5
  • ...and 74 more