Braid Group Representations arising from the Yang Baxter Equation

TL;DR

This paper investigates the representations of the braid group derived from the Yang Baxter equation with a specific focus on 4x4 solutions, showing that under certain conditions, the images are finite and relate to known link invariants.

Contribution

It characterizes the images of braid group representations from the Yang Baxter equation for 4x4 matrices and identifies conditions for obtaining link invariants and their relation to known algebraic structures.

Findings

Images are finite when eigenvalues are roots of unity.

Certain representations lead to known link invariants.

Group algebra sometimes factors through known algebras.

Abstract

This paper aims to determine the images of the braid group under representations afforded by the Yang Baxter equation when the solution is a nontrivial $4 \times 4$ matrix. Making the assumption that all the eigenvalues of the Yang Baxter solution are roots of unity, leads to the conclusion that all the images are finite. Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as particular specializations of a known invariant.