Handle decompositions of rational homology balls and Casson–Gordon invariants

Aceto, Paolo; Golla, Marco; Lecuona, Ana G.

doi:10.1090/proc/14035

Cited by 8 publications

(13 citation statements)

References 13 publications

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“…LetD ⊆ G 0 denote a disk which contains the 4 feet of the two tubes, and also intersects the dividing set of G 0 in a single arc. We may pickD to consist of the product of a subarc of K 0 , containing the 4 feet of the tubes, and a sub-interval of [0,1]. Let γ be a path inD connecting a foot of one tube to a foot of the other tube.…”

Section: A Generalized Doubling Relationmentioning

confidence: 99%

“…Part (ii): The homomorphism torsion order can be rephrased as the minimum N such that if i and j are non-negative integers with i + j = N , then there is a rank 1, free submodule F ⊆ HFK − (K) such that u i v j • HFK − (K) ⊆ F . For an L-space knot K, the minimal such N is easily seen to be 1…”

Section: Computations Of Generalized Torsion Ordersmentioning

confidence: 99%

“…Mizuma [21, Theorem 1.5] showed that if

K

is a ribbon knot which has Alexander polynomial 1 and whose Jones polynomial has non‐vanishing derivative at

t = - 1

, then

K

has fusion number at least 3. More recently, Aceto, Golla, and Lecuona [1, Corollary 2.3] have given obstructions using the Casson–Gordon signature invariants of

Σ (K)

. Note that these bounds do not give useful information for the ribbon knots

K = T_{p, q} # {\bar{T}}_{p, q}

for odd

p

and

q

since they involve

H_{1} false(normalΣ (K) false)

, and

Σ (T_{p, q} # {\bar{T}}_{p, q})

is the connected sum of the Brieskorn spheres

Σ (2, p, q)

and

- Σ (2, p, q)

.…”

Section: Introductionmentioning

confidence: 99%

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Knot cobordisms, bridge index, and torsion in Floer homology

Juhász

Miller

Zemke

2020

Journal of Topology

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Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for Tp,q and Tp,q#T p,q , respectively. We also show that the bridge index of Tp,q is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show that knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot.

show abstract

Section: A Generalized Doubling Relationmentioning

confidence: 99%

Section: Computations Of Generalized Torsion Ordersmentioning

confidence: 99%

“…Mizuma [21, Theorem 1.5] showed that if

K

is a ribbon knot which has Alexander polynomial 1 and whose Jones polynomial has non‐vanishing derivative at

t = - 1

, then

K

has fusion number at least 3. More recently, Aceto, Golla, and Lecuona [1, Corollary 2.3] have given obstructions using the Casson–Gordon signature invariants of

Σ (K)

. Note that these bounds do not give useful information for the ribbon knots

K = T_{p, q} # {\bar{T}}_{p, q}

for odd

p

and

q

since they involve

H_{1} false(normalΣ (K) false)

, and

Σ (T_{p, q} # {\bar{T}}_{p, q})

is the connected sum of the Brieskorn spheres

Σ (2, p, q)

and

- Σ (2, p, q)

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Knot cobordisms, bridge index, and torsion in Floer homology

Juhász

Miller

Zemke

2020

Journal of Topology

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show abstract

“…Papers that touch on aspects of these topics include [1, 2, 4, 6, 11, 13, 16-19, 22-24, 30, 32]. Through the use of cyclic branched covers, this study is related to the study of the handlebody structure of cobordisms between three-manifold, as presented, for instance, in [2].…”

Section: Introductionmentioning

confidence: 99%

Critical point counts in knot cobordisms: abelian and metacyclic invariants

Livingston¹

2021

Preprint

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For a pair of knots K 1 and K 0 , we consider the set of four-tuples of integers (g, c 0 , c 1 , c 2 ) for which there is a cobordism from K 1 to K 0 of genus g having c i critical points of each index i. We describe basic properties that such sets must satisfy and then build homological obstructions to membership in the set. These obstructions are determined by knot invariants arising from cyclic and metacyclic covering spaces.

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“…There were several lower bounds on the fusion number prior to the torsion order. Most of them come from cyclic branched covers of knots [NN82,Kan10,AGL18]. For instance, given any knot K, the minimum number of generators for H 1 (Σ(K); Z) is a lower bound on 2F(K), where Σ(K) is the 2-fold cyclic branched cover of K [NN82, Proposition 2].…”

Section: Introductionmentioning

confidence: 99%

Ribbon knots, cabling, and handle decompositions

Hom

Kang

Park

2021

Mathematical Research Letters

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The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the (p, 1)-cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number p. Our main tools are Juhász-Miller-Zemke's bound on fusion number coming from the torsion order of knot Floer homology and Hanselman-Watson's cabling formula for immersed curves.

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Handle decompositions of rational homology balls and Casson–Gordon invariants

Cited by 8 publications

References 13 publications

Knot cobordisms, bridge index, and torsion in Floer homology

Knot cobordisms, bridge index, and torsion in Floer homology

Critical point counts in knot cobordisms: abelian and metacyclic invariants

Ribbon knots, cabling, and handle decompositions

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