Kauffman bracket skein algebra of the 4-holed disk

TL;DR

This paper constructs a monomial basis and provides a presentation for the Kauffman bracket skein algebra of a 4-holed disk, linking it to the SL(2,C)-character variety of a rank 4 free group.

Contribution

It introduces a new monomial basis and a presentation for the skein algebra of the 4-holed disk, based on insights into the character variety.

Findings

Established a monomial basis for the algebra.

Derived a presentation of the algebra.

Connected the algebra to the SL(2,C)-character variety.

Abstract

We give a monomial basis for the Kauffman bracket skein algebra of the $4$-holed disk, and find a presentation. This is based on an insight into the ${\rm SL}(2,\mathbb{C})$-character variety of the rank $4$ free group.