Cluster C*-algebras and knot polynomials

TL;DR

This paper constructs a representation of braid groups within a cluster C*-algebra derived from surface triangulations, linking algebraic K-theory to knot invariants like Jones and HOMFLY polynomials.

Contribution

It introduces a novel algebraic framework connecting cluster C*-algebras with topological knot invariants via surface triangulations.

Findings

Laurent polynomials in K-theory serve as topological invariants

Jones polynomial corresponds to sphere with two cusps

HOMFLY polynomial corresponds to torus with one cusp

Abstract

We construct a representation of the braid groups in a cluster C*-algebra coming from a triangulation of the Riemann surface S with one or two cusps. It is shown that the Laurent polynomials attached to the K-theory of such an algebra are topological invariants of the closure of braids. In particular, the Jones and HOMFLY polynomials of a knot correspond to the case S being a sphere with two cusps and a torus with one cusp, respectively.