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

Knot homology theories are powerful tools in low-dimensional topology, which can be developed from multiple viewpoints. The most common viewpoints are symplectic geometry and the representation theory of quantum groups, and unifying these perspectives is an ongoing project. In this talk, we'll demonstrate a simple relationship between knot Floer homology and the $\mathfrak{gl}(1|1)$ quantum tangle invariant, by using the Burau representation of the braid group as a bridge between them. (We’ll also explain each of these three pieces.) As an application we’ll discuss new, Floer-theoretic invariants of tangles, which are a work in progress with Patricia Sorya.