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

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/