Representations of the Kauffman bracket skein algebra II: punctured surfaces

TL;DR

This paper develops a method to construct irreducible representations of the Kauffman skein algebra for punctured surfaces, linking algebraic invariants to geometric structures, with implications for understanding surface invariants.

Contribution

It introduces an inverse construction associating invariants to irreducible representations of the skein algebra for punctured surfaces, expanding previous work.

Findings

Constructed a correspondence between invariants and irreducible representations.

Analyzed the algebraic structure of the Thurston intersection form.

Restricted to surfaces with punctures, with future work extending this.

Abstract

In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation that realizes these invariants. The current article is restricted to surfaces with at least one puncture, a condition that will be lifted in subsequent work of the authors that relies on this one. A step in the proof is of independent interest, and describes the algebraic structure of the Thurston intersection form on the space of integer weight systems for a train track.