A Bennequin-Type Inequality and Combinatorial Bounds

Collari, Carlo

doi:10.1307/mmj/1565402474

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…The combinatorial bound presented in Theorem 1.3 is analogous to the bounds presented in [6,16] (see also [1,20,25]) for the Rasmussen and Rasmussen-Beliakova-Wehrli invariants. A possible direction of work might be to find an analogue of the combinatorial bound presented in [12]. Let us leave this matter aside for now, and let us turn to the last result of this section.…”

Section: 3mentioning

confidence: 99%

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Slice-torus Concordance Invariants and Whitehead Doubles of Links

Cavallo¹,

Collari²

2019

Can. J. Math.-J. Can. Math.

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In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants, and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent from the corresponding slice-torus link invariant.2010 Mathematics Subject Classification. 57M27. 1 These function were originally defined only for integer-valued slice-torus invariants. Of course, the same definition works for all slice-torus invariants, and most of the properties proved in [23] still hold. 1 2 ALBERTO CAVALLO AND CARLO COLLARIwhere ν is a slice-torus invariant and W ± (K, t) denotes the positive (resp. negative) ttwisted Whitehead double of K. These functions are non-increasing, non-constant, take values respectively in [0, 1] and [−1, 0], and assume both the maximal and the minimal possible values. In particular, if the slice-torus invariant is integer-valued all the information contained in each function can be condensed into a single integer. These integers, denoted by t ν and t ν , are defined as the maximal value of t such that F ν (K, t) and F ν (K, t), respectively, assume their maximum. It is not difficult to see that t ν (K) = −t ν (−K * ) − 1, where −K * is the mirror image of K with the orientation reversed, so these invariants contain the same amount of information. At the time of writing it is still unknown whether the invariant t ν can provide new information with respect to ν. In fact there are some hints in the opposite direction; for instance, it is known that t τ = 2τ − 1 ([15, Theorem 1.5]) and it has been conjectured that t s/2 = 3s/2 − 1 ([29]).The aim of the present paper is to extend these definitions and constructions to the case of links, and to describe some applications and examples. Before stating the main results of this paper let us recall a few basic facts about link concordance. The first thing one should point out is that the definition of concordance is no longer unique. Two oriented links in S 3 are said to be ⊲ weakly concordant if there exists a genus 0 connected, compact, oriented surface, properly embedded in S 3 × [0, 1], bounding the two links; ⊲ strongly concordant if there exists a disjoint union of annuli, properly embedded in S 3 × [0, 1], such that each of them bounds a component of each link. In particular, strongly concordant links should have the same number of components. A link is said to be weakly (resp. strongly) slice if is weakly (resp. strongly) concordant to an unlink. Similarly, one can define a (weak ) slice genus and a strong slice genus. The former is just the minimal genus of any connected, compact, oriented surface properly embedded in S 3 × [0, 1] bounding the link. The latter has a similar definition but one has t...

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Section: 3mentioning

confidence: 99%

“…Let D be an oriented link diagram representing the oriented link L. It can be easily shown (see, for instance, [12,Proposition 11] and subsequent proof) that…”

Section: 2mentioning

confidence: 99%

Slice-torus Concordance Invariants and Whitehead Doubles of Links

Cavallo¹,

Collari²

2019

Can. J. Math.-J. Can. Math.

Self Cite

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A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class

Sano

2020

J. Knot Theory Ramifications

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We give a description of Rasmussen’s [Formula: see text]-invariant from the divisibility of Lee’s canonical class. More precisely, given any link diagram [Formula: see text], for any choice of an integral domain [Formula: see text] and a non-zero, non-invertible element [Formula: see text], we define the [Formula: see text]-divisibility [Formula: see text] of Lee’s canonical class of [Formula: see text], and prove that a combination of [Formula: see text] and some elementary properties of [Formula: see text] yields a link invariant [Formula: see text]. Each [Formula: see text] possesses properties similar to [Formula: see text], which in particular reproves the Milnor conjecture. If we restrict to knots and take [Formula: see text], then our invariant coincides with [Formula: see text].

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A Bennequin-Type Inequality and Combinatorial Bounds

Cited by 2 publications

References 26 publications

Slice-torus Concordance Invariants and Whitehead Doubles of Links

Slice-torus Concordance Invariants and Whitehead Doubles of Links

A description of Rasmussen’s invariant from the divisibility of Lee’s canonical class

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