Abstract

Link Floer homology categorifies the ​multivariable Alexander polynomial, a classical invariant for knots and links. Motivated by constructions in Khovanov homology, one can ask if it is possible to define this invariant “locally”, i.e. to generalise it to tangles. ​A simpler question to start with is, of course: What is the Alexander polynomial of a tangle? As it turns out, this is not entirely clear.​

​There are several (a priori different) constructions to which I am going to add yet another one: In this talk, we consider a polynomial tangle invariant defined via generalised Kauffman states and Alexander codes. We will see that​ this invariant enjoys ​many properties of the classical​ multivariate ​Alexander polynomial, in particular invariance under mutation. We will then see how to interpret the tangle invariant geometrically. Finally, I will talk about how to make the transition to the Heegaard Floer world in the hope of defining a categorified version and, if time permits, ideas to make this construction glueable​.