Genus One Cobordisms Between Torus Knots

Feller, Peter; Park, JungHwan

doi:10.1093/imrn/rnaa027

Cited by 4 publications

(4 citation statements)

References 37 publications

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“…Firstly, when determining Gordian distance 1 pairs of torus knots, we relied on ν + , but speculated that the proof could be done using the Tristram-Levine signatures [Tri69,Lev69]; see [FP19,Remark 4•3]. Here, we partially confirm this speculation using signature calculations via the Hirzebruch-Brieskorn formula [Bri66,GG05].…”

Section: Introductionmentioning

confidence: 62%

“…This was done by comparing explicit constructions of cobordisms with the lower bound for the cobordism distance using the ν + -invariant [HW16] from the Heegaard Floer knot complex. As an application, the authors determined which pairs of torus knots have Gordian distance 1; see [FP19,Corollary 1.3]. The first result of this paper has two motivations.…”

Section: Introductionmentioning

confidence: 95%

“…We comment on the condition g 4 (K ) = 1. The point is that g 4 (K ) = 1 implies that {T p,q , T p ,q } must be one of an infinite family of pairs of torus knots, among which we know which are Gordian distance 1 apart, and, in fact, infinitely many are further apart [FP19]; see Section 3. With this the first part of Theorem 1•1 reduces to showing g top 4 (# n K ) ≥ n and c top 4 (# n K ) ≥ 2n for an explicit infinite family of knots.…”

Section: Introductionmentioning

confidence: 99%

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A note on the four-dimensional clasp number of knots

Feller¹,

Park²

2021

Math. Proc. Camb. Phil. Soc.

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Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterise those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number at least 2n. Our proof works in the topological category. To contrast this, we build a family of topologically slice knots for which the n-fold connected self-sum has 4-ball genus n and 4-dimensional clasp number at least 2n.

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Section: Introductionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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A note on the four-dimensional clasp number of knots

Feller¹,

Park²

2021

Math. Proc. Camb. Phil. Soc.

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“…For instance, for distinct positive parameters, Litherland [15] proved that the torus knots T (p, q) and T (p , q ) are linearly independent. On the other hand, Feller-Park [6] analyzed the question of determining for which pairs d(T (p, q), T (p , q )) = 1, resolving all but one case: d(T (3, 14), T (5, 8)). Related work includes [1,5] The triangle inequality states that for all K and J, d(K, J) ≤ g 4 (K) + g 4 (J).…”

Section: Introductionmentioning

confidence: 99%

The cobordism distance between a knot and its reverse

Livingston¹

2020

Preprint

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We consider the question of how knots and their reverses are related in the concordance group C. There are examples of knots for which K = K r ∈ C. This paper studies the cobordism distance, d(K, K r ). If K = K r ∈ C, then d(K, K r ) > 0 and it elementary to see that for all K, d(K, K r ) ≤ 2g 4 (K). It is known that d(K, K r ) can be arbitrarily large. Here we present a proof that for non-slice knots satisfying g 3 (K) = g 4 (K), one has d(K, K r ) ≤ 2g 4 (K) − 1. This family includes all strongly quasi-positive knots and all non-slice genus one knots. We also construct knots K of arbitrary four-genus for which d(K, K r ) = g 4 (K). Finding knots for which d(K, K r ) > g 4 (K) remains an open problem.

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The cobordism distance between a knot and its reverse

Livingston

2022

Proc. Amer. Math. Soc.

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We consider the question of how knots and their reverses are related in the concordance group C \mathcal {C} . There are examples of knots for which K ≠ K r ∈ C K \ne K^r \in \mathcal {C} . This paper studies the cobordism distance, d ( K , K r ) d(K, K^r) . If K ≠ K r ∈ C K \ne K^r \in \mathcal {C} , then d ( K , K r ) > 0 d(K, K^r) >0 and it is elementary to see that for all K K , d ( K , K r ) ≤ 2 g 4 ( K ) d(K, K^r) \le 2g_4(K) , where g 4 ( K ) g_4(K) denotes the four-genus. Here we present a proof that for non-slice knots satisfying g 3 ( K ) = g 4 ( K ) g_3(K) = g_4(K) , one has d ( K , K r ) ≤ 2 g 4 ( K ) − 1 d(K,K^r) \le 2g_4(K) -1 . This family includes all strongly quasi-positive knots and all non-slice genus one knots. We also construct knots K K of arbitrary four-genus for which d ( K , K r ) = g 4 ( K ) d(K,K^r) = g_4(K) . Finding knots for which d ( K , K r ) > g 4 ( K ) d(K,K^r) > g_4(K) remains an open problem.

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Genus One Cobordisms Between Torus Knots

Cited by 4 publications

References 37 publications

A note on the four-dimensional clasp number of knots

A note on the four-dimensional clasp number of knots

The cobordism distance between a knot and its reverse

The cobordism distance between a knot and its reverse

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