On skein algebras of planar surfaces

TL;DR

This paper studies the algebraic relations in skein algebras of planar surfaces, providing bounds on the degrees of generators of the defining relations depending on the invertibility of a specific element.

Contribution

It establishes degree bounds for the relations in skein algebras of planar surfaces, advancing understanding of their algebraic structure under different conditions.

Findings

Relations among generators are generated by relations of degree ≤6 for invertible cases.

Relations are generated by relations of degree ≤2k+2 for non-invertible cases.

Progress towards solving a problem in Kirby's list regarding skein algebra presentations.

Abstract

Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$. Let $\mathcal{S}_n$ denote the Kauffman bracket skein algebra of the $n$-holed disk $\Sigma_{0,n+1}$ over $R$. When $q+q^{-1}$ is invertible, in 2000 Przytycki and Sikora found a set of $n+{n\choose 2}+{n\choose 3}$ generators for $\mathcal{S}_n$; we show that the ideal of defining relations among these generators is generated by relations of degree $\le6$ supported by certain subsurfaces diffeomorphic to $\Sigma_{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, a set of $2^n-1$ generators for $\mathcal{S}_n$ was known to Bullock in 1999; we show that the ideal of defining relations is generated by relations of degree $\le 2k+2$ supported by certain subsurfaces diffeomorphic to $\Sigma_{0,k+1}$ with $k\le n$. These results are substantial progresses towards answering Problem 1.92 (J) in the Kirby's list.