On symmetries of peculiar modules, or $\delta$-graded link Floer homology is mutation invariant

Zibrowius, Claudius

doi:10.4171/jems/1201

Cited by 3 publications

(6 citation statements)

References 17 publications

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“…In the context of Heegaard Floer homology, analogous properties are known for the tangle invariant HFT(T ) [11]. Note, however, that components of the invariant BN(T ) are known to be much more complicated even when restricting to only compact components.…”

Section: Review Of the Tangle Invariants Kh And Bnmentioning

confidence: 80%

Khovanov homology and strong inversions

2022

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There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author (Adv. Math. 313 (2017), 915-946). In particular, we provide a counterexample to Conjecture 29 therein, as well as a refinement of and additional evidence for Conjecture 28.The Brieskorn spheres (2, q, 2nq ∓ 1) may be obtained by Dehn surgery on a torus knot in the three-sphere, namely, these are the integer homology spheres S 3 ±1/n (T 2,q ), where T 2,q is the positive (2, q) torus knot. These homology spheres admit Seifert fibrations, with base orbifold S 2 (2, q, 2nq ∓ 1). Denoting by (A, b) the two-fold branched cover of A with branch set b, each of these manifolds admits two descriptions as a two-fold branched cover:. This construction might be best termed as classical; for our purposes it is helpful to review the notation introduced in [9]. The first of these two-fold branched covers results from an involution on the Seifert fibred space that preserves an orientation on the fibres -we refer to this as the Seifert involution. The second of these arises from the Montesinos involution, which reverses an orientation on the fibres; the branch set in question arises from the Montesinos trick, that is, by first constructing a tangle (B 3 , τ ) over which the exterior of T 2,q is realized as a two-fold branched cover. We review the construction below as it is central to our enumeration of Kotelskiy is supported by an AMS-Simons travel grant. Watson is supported by an NSERC discovery/accelerator grant. Zibrowius is supported by the Emmy Noether Programme of the DFG, Project number 412851057. MSC2020: 57K10, 57K18.

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Section: Review Of the Tangle Invariants Kh And Bnmentioning

confidence: 80%

Khovanov homology and strong inversions

2022

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“…Isotoping the diagram so that the tangles are alternating, the mutated diagram then has at most 4 bad domains. Using the result of [11,Theorem 0.1] then the corollary follows as before.…”

Section: Montesinos Knotsmentioning

confidence: 91%

“…The formula (1-1) can be used in a further way: by a recent result of Zibrowius [11], mutation does not change HFK δ (K ), and hence leaves th(K ) unchanged.…”

Section: András I Stipsicz and Zoltán Szabómentioning

confidence: 99%

A note on thickness of knots

Stipsicz

Szabó

2022

Open Book Series

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We introduce a numerical invariant β(K ) ∈ ‫ގ‬ ∪ {0} of a knot K ⊂ S 3 which measures how nonalternating K is. We prove an inequality between β(K ) and the (knot Floer) thickness th(K ) of a knot K. As an application we show that all Montesinos knots have thickness at most one.Clearly, the diagram D is alternating if and only if B(D) = 0. Indeed, by takingwe get a knot invariant, which satisfies β(K ) = 0 if and only if K is an alternating knot. As it is typical for knot invariants given by minima of quantities over all diagrams, it is easy to find an upper bound on β(K ) (by determining B(D) for a diagram of K ), but it is harder to actually compute its value.MSC2020: 57K10.

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“…The invariant HFT(T ) can be also defined via Zarev's bordered sutured Heegaard Floer theory [Zar09]. In this alternate construction, the curved chain complex CFT ∂ (T ) is replaced by an a posteriori equivalent algebraic object, namely the bordered sutured type D structure associated with the tangle complement, which is equipped with a certain bordered sutured structure; see [Zib23b,Section 5]. This perspective gives rise to the following gluing result which relates the invariant HFT to link Floer homology HFL via Lagrangian Floer homology HF.…”

Section: A Gluing Theorem For Hftmentioning

confidence: 99%

“…We review some properties of the immersed curve invariant of Conway tangles due to the third author [Zib20]; see also [Zib19, Zib23b].…”

Section: The Tangle Invariantmentioning

confidence: 99%

Thin links and Conway spheres

Kotelskiy,

Watson,

Zibrowius

2024

Compositio Math.

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When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$ -grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname {HFT}$ and the Khovanov invariant $\widetilde {\operatorname {Kh}}$ that were developed by the authors in previous works.

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On symmetries of peculiar modules, or $\delta$-graded link Floer homology is mutation invariant

Cited by 3 publications

References 17 publications

Khovanov homology and strong inversions

Khovanov homology and strong inversions

A note on thickness of knots

Thin links and Conway spheres

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