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 4ended 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 $\widehat{\operatorname{HFT}}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\widehat{\operatorname{HFT}}$ and prove symmetry relations for $\widehat{\operatorname{HFT}}$ of 4ended tangles that echo those for $\nabla_T^s$.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1601.04915
 Bibcode:
 2016arXiv160104915Z
 Keywords:

 Mathematics  Geometric Topology
 EPrint:
 version accepted for publication in Algebraic and Geometric Topology