Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies

Moore, Allison H.; Starkston, Laura

doi:10.2140/agt.2015.15.43

Cited by 7 publications

(10 citation statements)

References 24 publications

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“…Since the knots K p,q are ribbon, this result is a special case of a more general observation pertaining to the behavior of knot Floer homology for families of ribbon knots related by twisting the ribbon disk. We outline the proof below; a complete proof of Theorem 7 is contained in Hedden and Watson (see the forthcoming paper, 'On the geography and botany of knot Floer homology') (see also [8,Lemma 12]).…”

Section: Kanenobu's Knotsmentioning

confidence: 99%

“…On the other hand, the chosen genus 4 Heegaard splitting (and associated presentation for the fundamental group) generalizes in a straightforward manner to the symmetric union of any pair of twist knots (note that K p,q is the symmetric union of figure-eight knots) [3,5]. These symmetric unions give a natural extension of the class of Kanenobu knots, and, in particular, the various homological invariants within a given family are identical (see [8,21], as well as the forthcoming paper of Hedden and Watson). Thus, a version of Proposition 1 applies to these knots, yielding further infinite families to which the techniques of this paper should apply to produce examples of hyperbolic, thin, non-QA knots with identical homological invariants.…”

Section: Closing Remarksmentioning

confidence: 99%

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Turaev torsion, definite 4-manifolds, and quasi-alternating knots

Greene

Watson²

2013

Bulletin of the London Mathematical Society

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We construct an infinite family of hyperbolic, homologically thin knots that are not quasi‐alternating. To establish the latter, we argue that the branched double‐cover of each knot in the family does not bound a negative‐definite 4‐manifold with trivial first homology and bounded second Betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.

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Section: Kanenobu's Knotsmentioning

confidence: 99%

Section: Closing Remarksmentioning

confidence: 99%

Turaev torsion, definite 4-manifolds, and quasi-alternating knots

Greene

Watson²

2013

Bulletin of the London Mathematical Society

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“…The arrow marked with ·u is the boundary map and it raises the homological grading by 1. Though computations similar to Lemma 10 can be found in [MS15,Sta12], for concreteness we provide a proof.…”

Section: Proof Of Main Theoremmentioning

confidence: 94%

Symmetric Unions Without Cosmetic Crossing Changes

Moore

2016

Association for Women in Mathematics Series

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A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic nonfibered three-bridge knots of constant determinant which satisfy the well-known cosmetic crossing conjecture. This conjecture asserts that the only crossing changes which preserve the isotopy type of a knot are nugatory.Definition 4. A symmetric union of J is an (unoriented) knot diagram obtained by replacing an elementary 0-tangle T 0 with an elementary n-tangle T n , with n = 0, ∞, along an axis of mirror symmetry in a diagram of J#m(J) as in Figure 2. A knot which admits a symmetric union diagram is called a symmetric union, and we denote a symmetric union of J by K n (J). The (unoriented) knot J is called the partial knot of K n (J), and K 0 (J) is J#m(J).The definition is due to Kinoshita and Teraksa [KT57]. Note that when J is oriented and n is even, K n (J) inherits an orientation from the connected sum of J with its reverse mirror image, but when n is odd, the orientation of K n (J) is not well-defined. To construct

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“…An interesting open question is the relationship between mutation and knot Floer homology. While many knot polynomials and homology theories are insensitive to mutation, the bigraded knot Floer homology groups can detect mutation [OS04c] and genus 2 mutation [MS15]. Conversely, explicit computations [BG12] and a combinatorial formulation [BL12] suggest that the δ-graded HFK groups are mutation-invariant.…”

Section: Introductionmentioning

confidence: 99%

Twisting, mutation and knot Floer homology

Lambert-Cole

2018

Quantum Topol.

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Let L be a knot with a fixed positive crossing and Ln the link obtained by replacing this crossing with n positive twists. We prove that the knot Floer homology HFK(Ln) 'stabilizes' as n goes to infinity. This categorifies a similar stabilization phenomenon of the Alexander polynomial. As an application, we construct an infinite family of prime, positive mutant knots with isomorphic bigraded knot Floer homology groups. Moreover, given any pair of positive mutants, we describe how to derive a corresponding infinite family of positive mutants with isomorphic bigraded HFK groups, Seifert genera, and concordance invariant τ .

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Genus-two mutant knots with the same dimension in knot Floer and Khovanov homologies

Cited by 7 publications

References 24 publications

Turaev torsion, definite 4-manifolds, and quasi-alternating knots

Turaev torsion, definite 4-manifolds, and quasi-alternating knots

Symmetric Unions Without Cosmetic Crossing Changes

Twisting, mutation and knot Floer homology

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