Knot Floer Homology, the Burau Representation, and Quantum gl(1 1)

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 $B - λI$, we define a Heegaard Floer homology theory that categorifies (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 gl(1 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 Uq(gl(1 1)) braid representation.