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

Joe Boninger (Boston College)

Abstract: The Burau representation of the braid group and its cousin, the Gassner representation of the pure braid group, have been studied for more than 80 years. Even so, questions remain about their faithfulness. Separately, these braid representations share a close connection with knot Floer homology through the Fox calculus. We'll explore the relationship between the Burau representation and knot Floer homology—ultimately, we'll show that if B is the full Burau matrix of any braid, then the determinant of any square submatrix of B - xI can be categorified by an appropriate Heegaard Floer theory. As an application, we'll demonstrate a simple correspondence between knot Floer homology and the gl(1|1) quantum tangle invariant. This contributes to a larger program of understanding knot Floer theory through the lens of quantum groups. No background knowledge on any of the titular topics is required for the talk.