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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
Zibrowius, Claudius 1
@article{10_2140_agt_2019_19_2233,
author = {Zibrowius, Claudius},
title = {Kauffman states and {Heegaard} diagrams for tangles},
journal = {Algebraic and Geometric Topology},
pages = {2233--2282},
publisher = {mathdoc},
volume = {19},
number = {5},
year = {2019},
doi = {10.2140/agt.2019.19.2233},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2233/}
}
TY - JOUR AU - Zibrowius, Claudius TI - Kauffman states and Heegaard diagrams for tangles JO - Algebraic and Geometric Topology PY - 2019 SP - 2233 EP - 2282 VL - 19 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2233/ DO - 10.2140/agt.2019.19.2233 ID - 10_2140_agt_2019_19_2233 ER -
Zibrowius, Claudius. Kauffman states and Heegaard diagrams for tangles. Algebraic and Geometric Topology, Tome 19 (2019) no. 5, pp. 2233-2282. doi: 10.2140/agt.2019.19.2233
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