Kauffman states and Heegaard diagrams for tangles

Abstract

We define polynomial tangle invariants ${\nabla }_T^s$ via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for ${\nabla }_T^s$ of $4$–ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants ${\nabla }_T^s$ can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of ${\nabla }_T^s$: a Heegaard Floer homology $\widehat{HFT}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\widehat{HFT}$ and prove symmetry relations for $\widehat{HFT}$ of $4$–ended tangles that echo those for ${\nabla }_T^s$.