The skein algebra of the Borromean rings complement

TL;DR

This paper explicitly computes the skein algebra of the Borromean rings complement, linking it to the character variety and providing a concrete algebraic description for this 3-manifold.

Contribution

It determines an explicit algebraic presentation of the skein algebra for the Borromean rings complement, a new result in the study of skein modules of 3-manifolds.

Findings

Explicit formula for the skein algebra of the Borromean rings complement

Isomorphism between the skein algebra and a quotient of a polynomial ring

Connection established between skein algebra and the SL_2(C)-character variety

Abstract

The skein algebra of an oriented $3$-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the $SL_2(\mathbb{C})$-character variety. In this paper we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in $S^3$.