Braids, Complex Volume, and Cluster Algebra

TL;DR

This paper explores a cluster algebra framework to interpret the complex volume of knots, constructing an R-operator from cluster mutations that corresponds to hyperbolic octahedra and relates cluster variables to edge parameters in volume calculations.

Contribution

It introduces a novel cluster algebraic approach to understanding complex volumes of knots, linking cluster mutations to hyperbolic geometric structures.

Findings

R-operator constructed from cluster mutations acts as a hyperbolic octahedron

Cluster variables correspond to edge parameters in complex volume computation

Provides a new algebraic perspective on knot invariants

Abstract

We try to give a cluster algebraic interpretation of complex volume of knots. We construct the R-operator from the cluster mutations, and we show that it is regarded as a hyperbolic octahedron. The cluster variables are interpreted as edge parameters used by Zickert in computing complex volume.