Kauffman bracket skein module of two families of Seifert manifolds

TL;DR

This paper computes the Kauffman bracket skein modules of specific Seifert manifolds, providing presentations and demonstrating their algebraic structures, including freeness and finite generation, with implications for link triviality.

Contribution

It offers explicit presentations of skein modules for two families of Seifert manifolds and analyzes their algebraic properties, including generators and triviality of the empty link.

Findings

Skein modules of ,1((k_1,1),(k_2,1)) are free with infinitely many generators for k_1,k_2 .

Skein modules of ,0((k_1,1),(k_2,1),(k_3,1)) are finitely generated for k_1,k_2,k_3 .

The empty link is not trivial in either case.

Abstract

We compute the Kauffman bracket skein modules of Seifert manifolds $\Sigma_{0,1}((k_1,1),(k_2,1))$ and $\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ by providing presentations of them. From the obtained presentations, we show that the Kauffman bracket skein modules of $\Sigma_{0,1}((k_1,1),(k_2,1))$ are free with infinitely many generators when $k_1,k_2\ge 1$ and that of $\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ are finitely generated when $k_1,k_2,k_3 \ge 2$. We also show that the empty link in either case is not trivial.