Slicing knots in definite 4-manifolds

Kjuchukova, Alexandra; Miller, Allison N.; Ray, Arunima; Sakallı, Sümeyra

doi:10.48550/arxiv.2112.14596

Cited by 3 publications

(2 citation statements)

References 33 publications

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“…Note that the techniques used in the proof of Theorem 1.1 cannot be used to obstruct a knot K with sd(K) = 1 from having tu(K) ≤ 2 since the algebraic invariants can differ at most by a factor of 2. It also seems unlikely that the Floer theoretic techniques of [İnc17] would alone be enough to answer Question 4.1 given the difficulty in obstructing knots from being H-slice in indefinite 4-manifolds [Kju+21]. Thus, new techniques are likely needed to answer the question above.…”

Section: Corollary 44 There Are Infinitely Many Knots {Kmentioning

confidence: 99%

Unknotting via null-homologous twists and multi-twists

Allen¹,

Ince²,

Kim³

et al. 2022

Preprint

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The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus 1.

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Section: Corollary 44 There Are Infinitely Many Knots {Kmentioning

confidence: 99%

Unknotting via null-homologous twists and multi-twists

Allen¹,

Ince²,

Kim³

et al. 2022

Preprint

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“…An important problem in the study of smooth 4-manifolds is to determine the minimal genus of an embedded surface representing a given homology class. The relative version of this problem for 4-manifolds with boundary has also received considerable attention [27,25,7,32,23,8,16,22,19,20,18].…”

Section: Introductionmentioning

confidence: 99%

On the slice genus of quasipositive knots in indefinite 4-manifolds

Baraglia¹

2022

Preprint

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Let X be a closed indefinite 4-manifold with b + (X) = 3 (mod 4) and with non-vanishing mod 2 Seiberg-Witten invariants. We prove a new lower bound on the genus of a properly embedded surface in X\B 4 representing a given homology class and with boundary a quasipositive knot K ⊂ S 3 . In the null-homologous case our inequality implies that the minimal genus of such a surface is equal to the slice genus of K. If X is symplectic then our lower bound differs from the minimal genus by at most 1 for any homology class that can be represented by a symplectic surface. Along the way, we also prove an extension of the adjunction inequality for closed 4-manifolds to classes of negative self-intersection without requiring X to be of simple type.

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