Virtual Braids and Cluster Algebras

TL;DR

This paper extends a cluster algebra-based braid group representation to virtual braids, revealing new algebraic properties and constructing related representations for flat braid groups, with implications for knot invariants.

Contribution

It introduces a novel representation of the virtual braid group derived from cluster algebras, demonstrating the failure of forbidden relations and expanding to flat braid groups.

Findings

Forbidden relations do not hold in the virtual braid group representation.

Constructed representations for flat and flat virtual braid groups.

Provides a new algebraic framework linking cluster algebras and virtual braids.

Abstract

In 2015 Hikami and Inoue constructed a representation of the braid group in terms of cluster algebra associated with the decomposition of the complement of the corresponding knot into ideal hyperbolic tetrahedra. This representation leads to the calculation of the hyperbolic volume of the complement of the knot that is the closure of the corresponding braid. In this paper, based on the Hikami-Inoue representation discussed above, we construct a representation for the virtual braid group. We show that the so-called "forbidden relations" do not hold in the image of the resulting representation. In addition, based on the developed method, we construct representations for the flat braid group and the flat virtual braid group.