Quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">gl</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy="false">|</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> and tangle Floer homology

Ellis, Alexander P.; Petkova, Ina; Vértesi, Vera

doi:10.1016/j.aim.2019.04.023

Cited by 15 publications

(7 citation statements)

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“…In [PV14], Petkova and Vértesi define a theory with similar properties to Ozsváth and Szabó's, which they call tangle Floer homology. Ellis, Petkova, and Vértesi [EPV15] show that the dg algebras of [PV14] categorify a tensor product of irreducible representations of U q (gl(1|1)), each of which is V or V * except for one factor L(λ n+1 ) which is neither V nor V * . The Heegaard diagrams motivating these two theories are different; however, it would be interesting to see whether relationships between the theories exist, especially since both theories are known to compute knot Floer homology in some form.…”

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confidence: 99%

On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

Manion

2018

Quantum Topol.

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We relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of Uq(gl(1|1)). Specifically, we consider two subalgebras Cr(n, S) and C l (n, S) of Ozsváth-Szabó's algebra B(n, S), and identify their Grothendieck groups with tensor products of representations V and V * of Uq(gl(1|1)), where V is the vector representation. We identify the decategorifications of Ozsváth-Szabó's DA bimodules for elementary tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsváth-Szabó's theory and Viro's quantum relative A 1 of the Reshetikhin-Turaev functor based on Uq(gl(1|1)).

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confidence: 99%

On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

Manion

2018

Quantum Topol.

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“…They use a more general definition of tangles, namely two-sided ones. In [EPV15], they and Ellis show that the decategorification of their invariant agrees with Sartori's generalisation of the Alexander polynomials to two-sided tangles via the representation theory of U q (gl(1|1)) [Srt13]. Thus, Petkova and Vértesi's theory fits nicely into the Reshetikhin-Turaev framework [RT91], making it analogous to Khovanov's tangle invariant [Kh01].…”

Section: Similar Work By Other Peoplementioning

confidence: 53%

Kauffman states and Heegaard diagrams for tangles

Zibrowius

2019

Algebr. Geom. Topol.

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We define polynomial tangle invariants ∇ s T via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for ∇ s T of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants ∇ s T can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of ∇ s T : a Heegaard Floer homology HFT for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on HFT and prove symmetry relations for HFT of 4-ended tangles that echo those for ∇ s T .Kauffman states and Heegaard diagrams for tangles 3 of L and we call R the mutating tangle in this mutation. If L is oriented, we choose an orientation of L that agrees with the one for L outside of R. If this means that we need to reverse the orientation of the two open components of R, then we also reverse the orientation of all other components of R during the mutation; otherwise we do not change any orientation. For an alternative, but equivalent definition, see definition 3.3 and remark 3.4.Theorem 0.2 along with the glueing formula for ∇ s T gives rise to the following result. Corollary 0.4 (3.5) The multivariate Alexander polynomial is invariant under Conway mutation after identifying the variables corresponding to the two open strands of the mutating tangle.This result has long been known for the univariate Alexander polynomial, see for example [LM87, proposition 11], but I have been unable to find a corresponding result for the multivariate polynomial in the literature. The fact that mutation invariance follows so easily from the symmetry relations of theorem 0.2 suggests that ∇ T is well-suited for studying the "local behaviour" of the Alexander polynomial. The homological tangle invariant HFTHeegaard Floer homology theories were first defined by Ozsváth and Szabó in 2001 [OS01]. With an oriented, closed 3-dimensional manifold M , they associated a family of homological invariants, the simplest of which is denoted by HF(M). Given an oriented (null-homologous) knot or link L in M , Ozsváth and Szabó, and independently J. Rasmussen, then defined filtrations on the chain complexes which give rise to the respective flavours of knot and link Floer homology [OS03a, Ras03, OS05], the simplest of which is denoted by HFL(L). The Alexander polynomial can be recovered from these groups as the graded Euler characteristic.Given corollary 0.4, it is only natural to ask for a Heegaard-Floer theoretic categorification of ∇ s T . To this end, we define a homology theory HFT as follows: given a Heegaard diagram for a tangle T (see definition 4.1) along with a site s of T , we define a finitely generated Abelian group which comes with two gradings: a relative homological Z-grading and an Alexander grading, which is an additional relative Z-grading for each component of the tangle:CFT(T, s) = h∈Z ←homological grading a∈Z |T| ←Alexander grading CFT h (T, s, a).

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“…It is interesting to compare the ideas described in this paper to the combinatorial tangle Floer theory by Petkova and Vértesi and the algebraic tangle homology theory by Ozsváth and Szabó as well as their corresponding decategorifications in terms of the representation theory of

{scriptU}_{q} false(frakturgl (1 | 1) false)

. In fact, the definition of our generalised peculiar modules

{CFT}^{-}

is primarily inspired by the invariants from , see Remark .…”

Section: Introductionmentioning

confidence: 99%

Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

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With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

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Quantum gl1|1 and tangle Floer homology

Cited by 15 publications

References 40 publications

On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

Kauffman states and Heegaard diagrams for tangles

Peculiar modules for 4‐ended tangles

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