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

PATCH Seminar (joint with Bryn Mawr Haverford, Penn, and Swarthmore)

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 explore the relationship between the Burau representation and knot Floer homology, and show that a large number of matrices related to the Burau representation can be categorified by appropriate Heegaard Floer theories. As an application, we 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.

In the morning background talk (at 9:30am in room KINSC E309), I will discuss Fox Calculus and Heegaard diagrams. Knot Floer homology, and related Heegaard Floer theories, are powerful tools with a variety of applications in low-dimensional topology. In his paper defining knot Floer homology, Rasmussen remarks that the theory is “essentially just a geometric realization of the Fox calculus.” We’ll explore the meaning of this observation, with the goal of understanding how knot Floer theory fits into the context of classical three-dimensional topology and knot theory. No background knowledge of the theory will be needed.