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… 
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3 Citations

3 Citations

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References

SHOWING 1-10 OF 55 REFERENCES
Alexander invariants of ribbon tangles and planar algebras
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
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
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 $S^3$, $D^3$…
The duality conjecture in formal knot theory
This paper deals with the formal knot theory of Kauffman [2]. In the first three sections we prove his Duality Conjecture ([2], p. 57); in the fourth we give a new and simpler proof of Kauffman's key…
Alexander-Conway invariants of tangles
We consider an algebra of (classical or virtual) tangles over an ordered circuit operad and introduce Conway-type invariants of tangles which respect this algebraic structure. The resulting…
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