Kauffman brackets, character varieties, and triangulations of surfaces

TL;DR

This paper explores the relationship between Kauffman brackets, skein algebras, and character varieties of surfaces, establishing a correspondence between algebraic representations and geometric points with weights.

Contribution

It demonstrates how irreducible skein algebra representations relate to points on the surface's character variety and vice versa, linking algebraic and geometric structures.

Findings

Irreducible skein algebra representations specify points on the character variety.

Each character variety point with weights determines a unique Kauffman bracket.

The work bridges algebraic invariants with geometric surface structures.

Abstract

A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We show how an irreducible representation of the skein algebra usually specifies a point of the character variety of homomorphisms from the fundamental group of the surface to PSL_2(C), as well as certain weights associated to the punctures of the surface. Conversely, we sketch a proof of the fact that each point of the character variety, endowed with appropriate puncture weights, uniquely determines a Kauffman bracket. Details will appear elsewhere.