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 $\widehat{\operatorname{HFT}}$ 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 4-ended tangles that echo those for $\nabla_T^s$.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2016
- DOI:
- 10.48550/arXiv.1601.04915
- arXiv:
- arXiv:1601.04915
- Bibcode:
- 2016arXiv160104915Z
- Keywords:
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- Mathematics - Geometric Topology
- E-Print:
- version accepted for publication in Algebraic and Geometric Topology