Kauffman states and Heegaard diagrams for tangles

Zibrowius, Claudius

doi:10.2140/agt.2019.19.2233

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Kauffman states and Heegaard diagrams for tangles

Claudius Bodo Zibrowius

Algebraic & Geometric Topology 19 (2019) 2233–2282

DOI: 10.2140/agt.2019.19.2233

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 $\hat{HFT}$ for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on $\hat{HFT}$ and prove symmetry relations for $\hat{HFT}$ of $4$ –ended tangles that echo those for $\nabla_{T}^{s}$ .

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Keywords

tangles, Alexander polynomial, Heegaard Floer homology, Conway mutation

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27

References

Publication

Received: 15 October 2017

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/

Volume 19, issue 5 (2019)