#### Volume 19, issue 5 (2019)

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Kauffman states and Heegaard diagrams for tangles

### Claudius Bodo Zibrowius

Algebraic & Geometric Topology 19 (2019) 2233–2282
##### 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 $\stackrel{̂}{HFT}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\stackrel{̂}{HFT}$ and prove symmetry relations for $\stackrel{̂}{HFT}$ of $4$–ended tangles that echo those for ${\nabla }_{T}^{s}$.

##### Keywords
tangles, Alexander polynomial, Heegaard Floer homology, Conway mutation
Primary: 57M25
Secondary: 57M27
##### Publication
Revised: 2 August 2018
Accepted: 6 September 2018
Published: 20 October 2019
##### Authors
 Claudius Bodo Zibrowius Department of Mathematics The University of British Columbia Vancouver, BC Canada https://cbz20.raspberryip.com/