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

Greene, Joshua Evan; Watson, Liam

doi:10.1112/blms/bds096

Cited by 23 publications

(43 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Greene conjectured that there exist only finitely many quasi-alternating links with a given determinant [Gre10, Conjecture 3.1]. We suspect that the present examples, like the Kanenobu knots and pretzel knots, also fail to be quasi-alternating, and that an argument similar to that made by Greene and Watson in [GW13] for the case of the Kanenobu knots can be made.…”

Section: +1supporting

confidence: 54%

Symmetric Unions Without Cosmetic Crossing Changes

Moore

2016

Association for Women in Mathematics Series

View full text Add to dashboard Cite

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

show abstract

Section: +1supporting

confidence: 54%

Symmetric Unions Without Cosmetic Crossing Changes

Moore

2016

Association for Women in Mathematics Series

View full text Add to dashboard Cite

show abstract

“…Later, a result by Greene and Watson (cf. [GW11]) and a result by Hedden and Watson (cf. [HW14]) allowed to find infinite families of generalised Kanenobu knots which additionally share the odd-Khovanov and the knot Floer homologies.…”

Section: Introductionmentioning

confidence: 72%

On d-invariants and generalized Kanenobu knots

Marengon

2016

J. Knot Theory Ramifications

View full text Add to dashboard Cite

We prove that for particular infinite families of L-spaces, arising as branched double covers, the d-invariants defined by Ozsváth and Szabó are arbitrarily large and small. As a consequence, we generalise a result by Greene and Watson by proving, for every odd number ∆ ≥ 5, the existence of infinitely many nonquasi-alternating homologically thin knots with determinant ∆ 2 , and a result by Hoffman and Walsh concerning the existence of hyperbolic weight 1 manifolds that are not surgery on a knot in S 3 . * m.marengon13@imperial.ac.uk

show abstract

“…It provided a categorification of the Alexander polynomial. As in [6], J. Greene and L. Watson obtained the skein exact sequence in knot Floer homology and established the following result:…”

Section: The Khovanov Cohomology Of Kanenobu Knotsmentioning

confidence: 99%

“…Later K. Qazaqzeh and N. Chbili [19] showed that there were only finite quasi-alternating links in the family of Kanenobu knots. In this sense, the Kanenobu knots can be seen as the "most non-quasialternating" class of knots (See [5,6]). However, a formula for calculating cohomology groups of general Kanenobu knots remains unknown.…”

Section: Introductionmentioning

confidence: 99%

A Recursive Formula for the Khovanov Cohomology of Kanenobu Knots

Lei¹,

Zhang²

2017

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. Kanenobu has given infinite families of knots with the same HOMFLY polynomial invariant but distinct Alexander module structure. In this paper, we give a recursive formula for the Khovanov cohomology of all Kanenobu knots K(p, q), where p and q are integers. The result implies that the rank of the Khovanov cohomology of K(p, q) is an invariant of p + q. Our computation uses only the basic long exact sequence in knot homology and some results on homologically thin knots.

show abstract

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

Cited by 23 publications

References 15 publications

Symmetric Unions Without Cosmetic Crossing Changes

Symmetric Unions Without Cosmetic Crossing Changes

On d-invariants and generalized Kanenobu knots

A Recursive Formula for the Khovanov Cohomology of Kanenobu Knots

Contact Info

Product

Resources

About