Blanchfield forms and Gordian distance

Borodzik, Maciej; Friedl, Stefan; Powell, Mark

doi:10.2969/jmsj/06831047

Cited by 16 publications

(35 citation statements)

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“…Splitting number. As we anticipated, the slice-torus link invariants can be used to obtain a lower bound for the splitting number sp of a link ( [2]), which is sometimes called weak splitting number ( [5]). Let us recall its definition first.…”

Section: Applicationsmentioning

confidence: 99%

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: Applicationsmentioning

confidence: 99%

Slice-torus Concordance Invariants and Whitehead Doubles of Links

Cavallo¹,

Collari²

2019

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

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“…The required lower bound is provided by Lemma 2.4 for the links L5a1, L6a4, L7a1, L7a3, L7a4, L7a6, L8a1, L8a8, L8a9, L8a16, and L9aN for N ∈ {1, 3,4,8,9,18,20,21,22,25,26,27,35,38, 40, 42}.…”

Section: Examplesmentioning

confidence: 99%

“…Unlinking numbers and 4-ball crossing numbers of prime nonsplit links with 9 or fewer crossings. [15], and Borodzik-Friedl-Powell [3] leads to the complete Table 1 of unlinking numbers of nonsplit prime links with crossing number at most 9. We also show that for all links in Table 1, the unlinking number is equal to the 4-ball crossing number.…”

Section: Introductionmentioning

confidence: 99%

“…We are grateful to Maciej Borodzik, Stefan Friedl and Mark Powell for bringing the unknown values in Kohn's table to our attention in their paper [3]. The first author thanks the University of Glasgow for its hospitality.…”

mentioning

confidence: 96%

“…With the exception of the links in Examples 1.2 and 1.4, the unlinking numbers were known previously due to work of Kauffman-Taylor, Kohn, Kawauchi, and Borodzik-Friedl-Powell [3,[14][15][16], and in many cases these were also known to be equal to the 4-ball crossing number.…”

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confidence: 99%

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