Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$

Abstract

The Burau representation of braid groups and knot Floer homology share a link to the Fox calculus. We make this connection explicit, with the following outcome: if $B$ is the full Burau matrix of any braid, and $A$ is any square submatrix of $B - \lambda I$, we define a Heegaard Floer homology theory that categorifies $\det(A)$ and is an invariant of the braid. We also describe an analogous construction for the Gassner representation. Then, we leverage the relationship between the Burau representation and quantum $\mathfrak{gl}(1 \vert 1)$ to exhibit connections between the latter and Heegaard Floer homology. We associate a bordered sutured Heegaard Floer homology group to any tangle, and give a simple, geometric proof that our invariant recovers the $U_q(\mathfrak{gl}(1 \vert 1))$ braid representation.