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Biracks and Switch Braid Quivers

Max Chao-Haft, Sam Nelson

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.

Biracks and Switch Braid Quivers

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.

Paper Structure

This paper contains 9 sections, 15 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Biracks and Braids
  3. Switch Braid Representations
  4. Switch Structure on $X^n$
  5. When $\varphi_B$ is a Switch Automorphism
  6. Examples
  7. Switch Braid Quivers
  8. Switch Braid Quiver Cocycle Enhancements
  9. Questions

Key Result

Corollary 1

The birack counting invariant is an invariant of knots and links.

Theorems & Definitions (39)

  • Definition 1
  • Example 1
  • Definition 2
  • Corollary 1
  • Definition 3
  • Remark 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Theorem 4
  • ...and 29 more