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Presentations of Kauffman bracket skein algebras of planar surfaces

Haimiao Chen

Abstract

Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$, and suppose $q+q^{-1}$ is invertible in $R$. For each planar surface $Σ_{0,n+1}$, we present its Kauffman bracket skein algebra over $R$ by explicit generators and relations. The presentation is independent of $R$, and can be considered as a quantization of the trace algebra of $n$ generic $2\times 2$ unimodular matrices.

Presentations of Kauffman bracket skein algebras of planar surfaces

Abstract

Let be a commutative ring with identity and a fixed invertible element , and suppose is invertible in . For each planar surface , we present its Kauffman bracket skein algebra over by explicit generators and relations. The presentation is independent of , and can be considered as a quantization of the trace algebra of generic unimodular matrices.
Paper Structure (9 sections, 11 theorems, 91 equations, 10 figures)

This paper contains 9 sections, 11 theorems, 91 equations, 10 figures.

Table of Contents

  1. Introduction
  2. A revision of the classical result
  3. The defining ideal of relations
  4. Notations and techniques
  5. Commuting relations
  6. Quantization of classical relations of type II
  7. Quantization of classical relations of type I
  8. The presentation
  9. Proofs for some identities in Section \ref{['sec:presentation']}

Key Result

Proposition 3.2

For $(i_1,i_2,i_3,i_4)$ in cyclic order,

Figures (10)

  • Figure 1: Skein relations.
  • Figure 2: The local rule for defining the symbols $s_{i^\ast_1\cdots i^\ast_r}$.
  • Figure 3: (a): $s_{\hat{2}\hat{4}5}=t_{245}+\beta t_5t_{24}$; (b): $s_{2\hat{4}5}=s_{\hat{2}\hat{4}5}+\beta t_2s_{\hat{4}5}$; (c): $s_{245}=s_{2\hat{4}5}+\beta t_4s_{25}$.
  • Figure 4: (a): $s_{1345\overline{3}}$; (b): $s_{341\overline{4}}$; (c): $s_{23451\overline{3}}$.
  • Figure 5: Two simplified local relations.
  • ...and 5 more figures

Theorems & Definitions (26)

  • proof : Proof of (\ref{['eq:typeI']}) and (\ref{['eq:typeII']})
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 16 more