Positivities of Knots and Links and the Defect of Bennequin Inequality

Hamer, Jesse; Ito, Tetsuya; Kawamuro, Keiko

doi:10.1080/10586458.2019.1596848

Cited by 6 publications

(4 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is a conjecture (popularized by Hedden, Etnyre and Van Horn-Morris amongst others) that the Bennequin inequality is an equality if and only if K is strongly quasipositive; compare also [10]. As a consequence of Theorem D, we confirm this conjecture for knots with canonical genus g (the minimum genus of a canonical surface) equal to the genus.…”

Section: Corollary C Let Be a Seifert Surface That Is Isotopic To A Canonical Surface Then Is Quasipositive If And Only If Every Unknot Csupporting

confidence: 82%

“…Next we observe the following criterion for quasipositivity. This allows, see the example below, to determine the strong-quasipositivity status of all prime knots up to 13 crossings, in particular recovering the recently completed calculation [10] of the strong-quasipositivity status of prime knots up to 12 crossings. Throughout this subsection, let y denote a slice-torus invariant.…”

Section: Corollary C Let Be a Seifert Surface That Is Isotopic To A Canonical Surface Then Is Quasipositive If And Only If Every Unknot Cmentioning

confidence: 97%

See 1 more Smart Citation

Almost positive links are strongly quasipositive

2022

View full text Add to dashboard Cite

We prove that any link admitting a diagram with a single negative crossing is strongly quasipositive. This answers a question of Stoimenow’s in the (strong) positive. As a second main result, we give a simple and complete characterization of link diagrams with quasipositive canonical surface (the surface produced by Seifert’s algorithm). As applications, we determine which prime knots up to 13 crossings are strongly quasipositive, and we confirm the following conjecture for knots that have a canonical surface realizing their genus: a knot is strongly quasipositive if and only if the Bennequin inequality is an equality.

show abstract

Section: Corollary C Let Be a Seifert Surface That Is Isotopic To A Canonical Surface Then Is Quasipositive If And Only If Every Unknot Csupporting

confidence: 82%

Section: Corollary C Let Be a Seifert Surface That Is Isotopic To A Canonical Surface Then Is Quasipositive If And Only If Every Unknot Cmentioning

confidence: 97%

Almost positive links are strongly quasipositive

2022

View full text Add to dashboard Cite

show abstract

“…The concordance invariant τ (K) defined using Heegaard Floer homology [OS03] gives similar bounds [OS03,Pla04]: HIK19]). Let K be a knot type in S 3 .…”

Section: Introductionmentioning

confidence: 86%

Comparing Bennequin-type inequalities

Aceves,

Kawamuro,

Truong

2020

Preprint

Self Cite

View full text Add to dashboard Cite

The slice-Bennequin inequality states an upper bound for the self-linking number of a knot in terms of its four-ball genus. The s-Bennequin and τ -Bennequin inequalities provide upper bounds on the self-linking number of a knot in terms of the Rasmussen s invariant and the Ozsváth-Szabó τ invariant. We exhibit examples in which the difference between self-linking number and four-ball genus grows arbitrarily large, whereas the s-Bennequin inequality and the τ -Bennequin inequality are both sharp.

show abstract

“…As we have discussed in [HIK,Section 6], one can further deform the diagram into a closed braid diagram, preserving the disk and twisted band decomposition structure of S D . Since every twisted band has twisted in a positive direction, the closed braid obtained from S D is the closure strongly quasipositive braid (see [HIK,Theorem 6.4]. 2.3.…”

Section: Proof Of Propertiesmentioning

confidence: 99%

Successively almost positive links

Ito¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

As an extension of positive or almost positive diagrams and links, we introduce a notion of successively almost positive diagrams and links, and good successively almost positive diagrams and links. We review various properties of positive links or almost positive links, and explain how they can be extended to (good) successively almost positive links. Our investigation also leads to an improvement of known results of positive or almost positive links.

show abstract

Positivities of Knots and Links and the Defect of Bennequin Inequality

Cited by 6 publications

References 36 publications

Almost positive links are strongly quasipositive

Almost positive links are strongly quasipositive

Comparing Bennequin-type inequalities

Successively almost positive links

Contact Info

Product

Resources

About