Scissor equivalence for torus links

Baader, Sebastian

doi:10.1112/blms/bds044

Cited by 24 publications

(32 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We highlight our interest in the above theorem by noting again that algebraic knots, and torus knots in particular, are L-space knots. The minimal unknotting sequences of torus knots have recently attracted a lot of interest; see for example [1,2,9,23,35]. The Gordian distance between algebraic knots is closely related to studying adjacency of singularities; see [6,8].…”

Section: Introductionmentioning

confidence: 99%

The ϒ function ofL–space knots is a Legendre transform

Borodzik

Hedden

2017

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Given an L-space knot we show that its Υ function is the Legendre transform of a counting function equivalent to the d-invariants of its large surgeries. The unknotting obstruction obtained for the Υ function is, in the case of L-space knots, contained in the d-invariants of large surgeries. Generalizations apply for connected sums of L-space knots, which imply that the slice obstruction provided by Υ on the subgroup of concordance generated by L-space knots is no finer than that provided by the d-invariants.

show abstract

Section: Introductionmentioning

confidence: 99%

The ϒ function ofL–space knots is a Legendre transform

Borodzik

Hedden

2017

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…Here are three examples: S 1 = {0, 3, 5, 6, 8} → {(0, 4), (1,4), (1,2), (2, 2), (2, 1), (3, 1), (3, 1), (4, 1), (4, 0)}. Sequences of points constructed in this way are called staircases.…”

Section: Appendix a Staircases And The Upsilon Function For Torus Knotsmentioning

confidence: 99%

The four-genus of connected sums of torus knots

Livingston¹,

Cott²

2017

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

We study the four-genus of linear combinations of torus knots: g 4 (aT (p, q)# −bT (p , q )). Fixing positive p, q, p , and q , our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram-Levine signature function; for the second, the recently defined Upsilon function of Ozsváth-Stipsicz-Szabó determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.

show abstract

“…Non-isotopic torus knots have non-zero cobordism distance; in fact, non-trivial positive torus knots are linearly independent in the concordance group [Lit79]. Trotter's classical knot signature [Tro62] and the Tristram-Levine signatures [Tri69,Lev69] allow to determine the cobordism distance of most torus knots of two fixed braid indices up to a constant [Baa12]. The modern Heegaard Floer concordance invariants ν + [HW16] and Υ [OSS17] lead to better bounds on cobordism distance depending on the braid indices [BCG17,FK17].…”

Section: Introductionmentioning

confidence: 99%

Genus One Cobordisms Between Torus Knots

Feller

Park

2020

International Mathematics Research Notices

View full text Add to dashboard Cite

We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using ν + from the Heegaard Floer knot complex and explicit cobordisms constructions. As an application, we determine the induced subgraph of the Gordian graph on the set of vertices that are given by torus knots. Also, we determine the pairs of Thurston-Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.

show abstract

Scissor equivalence for torus links

Abstract: We show that the cobordism distance of torus links is determined by the profiles of their signature functions, up to a constant factor.

Cited by 24 publications

References 12 publications

The ϒ function ofL–space knots is a Legendre transform

The ϒ function ofL–space knots is a Legendre transform

The four-genus of connected sums of torus knots

Genus One Cobordisms Between Torus Knots

Contact Info

Product

Resources

About