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Coordinates for ${\rm SL}_3$-web basis elements in closed surfaces

Zhe Sun, Zhihao Wang

Abstract

The ${\rm SL}_3$-skein algebra of a closed surface $Σ_g$ is a quantization of the ${\rm SL}_3$ character variety of $Σ_g$, where $g$ denotes the genus of the surface. This algebra admits a basis consisting of non-elliptic web diagrams in $Σ_g$. In this paper, we introduce explicit coordinates for non-elliptic web diagrams on $Σ_g$, yielding a parametrization by a submonoid of $\mathbb Z^{d}$. Here $d = 16g - 16$ for $g \ge 2$ and $d = 4$ in the torus case $g = 1$, coinciding with the dimension of the corresponding character variety.

Coordinates for ${\rm SL}_3$-web basis elements in closed surfaces

Abstract

The -skein algebra of a closed surface is a quantization of the character variety of , where denotes the genus of the surface. This algebra admits a basis consisting of non-elliptic web diagrams in . In this paper, we introduce explicit coordinates for non-elliptic web diagrams on , yielding a parametrization by a submonoid of . Here for and in the torus case , coinciding with the dimension of the corresponding character variety.
Paper Structure (19 sections, 20 theorems, 121 equations, 24 figures)

This paper contains 19 sections, 20 theorems, 121 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. The ${\rm SL}_3$-skein algebra
  5. ${\rm SL}_3$-skein algebras
  6. Bases of the ${\rm SL}_3$-skein algebras
  7. Graded ${\rm SL}_3$-skein algebras
  8. Unbounded ${\rm SL}_3$-laminations and their shear coordinates
  9. Unbounded ${\rm SL}_3$-laminations
  10. ${\rm SL}_3$-shear coordinates
  11. General positions of basis elements in pants decompositions of a closed surface
  12. Webs in an annulus
  13. Braids and braided webs in the annulus
  14. Non-elliptic web diagrams in the annulus
  15. Twist numbers
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\{C_j\}_{1 \le j \le r}$ be an oriented pants decomposition of $\Sigma_g$ with $g \ge 2$, together with a dual graph $\Gamma$, where $r = 3g-3$. The coordinate map is injective. Moreover, where $\Theta$ is the submonoid of $\mathbb N^r \times \mathbb N^{r} \times \mathbb Z^r \times \mathbb Z^r \times \mathbb Z^{\mathbb P} \times \mathbb Z^{\mathbb P}$ defined in Definition def-image-theta.

Figures (24)

  • Figure 1: A flip move.
  • Figure 2: The left picture is an example for $\mathcal{H}_{-3}$ in $\mathbb D_3$, the right picture is an example for a corner arc in $\mathbb D_3$ (the orientation of the corner arc is arbitrary).
  • Figure 3: Clockwise direction of spiralling.
  • Figure 4: We depict $\Sigma_{0,3}$ as a pair of pants with its boundary removed. The triangulation is given by the three red curves. The set $I(\Sigma_{0,3})$ is labeled by $11,12,21,22,31,32,v,v'$, where $v$ (resp. $v'$) lies in the front (resp. back) triangle. The three punctures of $\Sigma_{0,3}$ (equivalently, the three boundary components of the pair of pants) are labeled by $C_1,C_2,C_3$.
  • Figure 5: The picture for $W \cap (P' \setminus\widetilde{P})$.
  • ...and 19 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Theorem 2.6: frohman20223, Proposition 4
  • Remark 2.7
  • Lemma 2.8
  • ...and 43 more