# Kauffman states and Heegaard diagrams for tangles

@article{Zibrowius2019KauffmanSA,
title={Kauffman states and Heegaard diagrams for tangles},
author={Claudius Zibrowius},
journal={Algebraic \& Geometric Topology},
year={2019}
}
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… 3 Citations ## Figures and Tables from this paper Peculiar modules for 4‐ended tangles With a 4‐ended tangle T , we associate a Heegaard Floer invariant CFT∂(T) , the peculiar module of T . Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, Thin links and Conway spheres • Mathematics • 2020 When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal δ-grading. This leads to the broader class of thin links that one would like to L-space knots have no essential Conway spheres • Mathematics • 2020 We prove that L-space knots do not have essential Conway spheres with the technology of peculiar modules, a Floer theoretic invariant for tangles. ## References SHOWING 1-10 OF 55 REFERENCES Alexander invariants of ribbon tangles and planar algebras • Mathematics Journal of the Mathematical Society of Japan • 2018 Ribbon tangles are proper embeddings of tori and cylinders in the$4$-ball~$B^4$, "bounding"$3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant$\mathsf{A}$of Twisting, mutation and knot Floer homology Let$\mathcal{L}$be a knot with a fixed positive crossing and$\mathcal{L}_n$the link obtained by replacing this crossing with$n$positive twists. We prove that the knot Floer homology The Alexander polynomial as quantum invariant of links In these notes we collect some results about finite-dimensional representations of$U_{q}(\mathfrak {gl}(1\mid1))$and related invariants of framed tangles, which are well-known to experts but On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology We relate decategorifications of Ozsv\'ath-Szab\'o's new bordered theory for knot Floer homology to representations of$\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras Bordered Floer homology for manifolds with torus boundary via immersed curves • Mathematics • 2016 This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If$M$is such a manifold, we show that the type D structure The Multivariable Alexander Polynomial on Tangles The Multivariable Alexander Polynomial on Tangles Jana Archibald Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2010 The multivariable Alexander polynomial (MVA) is a Floer homology and surface decompositions Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under Combinatorial tangle Floer homology • Mathematics • 2014 In this paper we extend the idea of bordered Floer homology to knots and links in$S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in$S^3$,$D^3\$
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