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Matrix Formulae and Skein Relations for Quasi-Cluster Algebras

Cody Gilbert, McCleary Philbin, Kayla Wright

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.

Matrix Formulae and Skein Relations for Quasi-Cluster Algebras

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.
Paper Structure (15 sections, 22 theorems, 28 equations, 28 figures)

This paper contains 15 sections, 22 theorems, 28 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Quasi-cluster algebras
  3. Expansion formula via snake and band graphs for non-orientable surfaces
  4. Coefficients using principal laminations
  5. Matrix formulae on orientable surfaces with respect to principal laminations
  6. Matchings of snake and band graphs
  7. The standard $\boldsymbol{M}$-path
  8. Example with closed curve
  9. Expansion formula for one-sided closed curves using matrix products
  10. Good matching enumerators for one-sided closed curves
  11. The standard $\boldsymbol{M}$-path for a one-sided closed curve
  12. Möbius band example
  13. Reflection and rotation example
  14. Skein relations for one-sided closed curves
  15. Appendix

Key Result

Theorem 3.6

Let $\gamma$ be a regular arc overlayed on a quasi-triangulation $T$ without quasi-arcs of a non-orientable $(\mathbf{S}, \mathbf{M})$. Let $\tilde{\gamma}$ be one of the two lifts of $\gamma$ on the orientable double cover $(\mathbf{S}, \mathbf{M})$. Let $\mathcal{G}_{\tilde{\gamma}}$ be the snake where cross$_T(\gamma) = x_{i_1} \cdots x_{i_d}$ is the crossing monomial which keeps track of the

Figures (28)

  • Figure 1: The skein relations used to define quasi-mutation (3).
  • Figure 2: A tile with the four cardinal directions.
  • Figure 3: Examples of the conditions for a laminations. On the left, the case where $\gamma$ is not a quasi-arc and on the right, the case where $\gamma$ is a quasi-arc.
  • Figure 4: The definition of $S$ and $Z$ intersections on the leftmost and middle figures.
  • Figure 5: Orientation of diagonals induced by a perfect matching of a snake graph.
  • ...and 23 more figures

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 53 more