Categorified invariants and the braid group

TL;DR

This paper explores two advanced categorified invariants derived from Khovanov and Heegaard Floer homologies, demonstrating their effectiveness in solving the word problem in braid groups but not the conjugacy problem.

Contribution

It introduces and analyzes two categorified braid invariants, showing their capabilities and limitations in solving fundamental problems in braid group theory.

Findings

Each invariant solves the word problem in the braid group.

Neither invariant solves the conjugacy problem.

The invariants are derived from Khovanov and Heegaard Floer homologies.

Abstract

We investigate two "categorified" braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer homology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group.