Biracks and Switch Braid Quivers

TL;DR

This paper introduces new algebraic structures called switches and biracks to define novel invariants for braids, leading to polynomial invariants through categorification and decategorification methods.

Contribution

It develops a switch structure on permutation representations of the braid group and constructs quiver-valued invariants, advancing the algebraic tools for braid analysis.

Findings

Defined switch automorphisms for braid representations

Constructed quiver-valued invariants of braids

Derived new polynomial invariants via decategorification

Abstract

We consider birack and switch colorings of braids. We define a switch structure on the set of permutation representations of the braid group and consider when such a representation is a switch automorphism. We define quiver-valued invariants of braids using finite switches and biracks and use these to categorify the birack 2-cocycle invariant for braids. We obtain new polynomial invariants of braids via decategorification of these quivers.