Chirurgies de Dehn de pente �1 sur certains n?uds dans lec 3-vari�t�s

Mathieu, Yves; Domergue, Michel

doi:10.1007/bf01456339

Cited by 5 publications

(4 citation statements)

References 7 publications

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“…The minimum number of generalized crossings one must add to the unknot to construct a given knot is called its untwisting number, and was first defined in [MD88]. The untwisting number of knots was investigated in [Liv02,CT14,İnc16,İnc17,McC19,McC21,Liv19].…”

Section: Background and Elementary Observationsmentioning

confidence: 99%

Slicing knots in definite 4-manifolds

Kjuchukova¹,

Miller²,

Ray³

et al. 2021

Preprint

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We study the CP 2 -slicing number of knots, i.e. the smallest m ≥ 0 such that a knot K ⊆ S 3 bounds a properly embedded, null-homologous disk in a punctured connected sum (# m CP 2 ) × . We give a lower bound on the smooth CP 2 -slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth CP 2slicing number. We also give an upper bound on the topological CP 2 -slicing number in terms of the Seifert form and find knots for which the smooth and topological CP 2 -slicing numbers are both finite, nonzero, and distinct.

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Section: Background and Elementary Observationsmentioning

confidence: 99%

Slicing knots in definite 4-manifolds

Kjuchukova¹,

Miller²,

Ray³

et al. 2021

Preprint

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“…In [MD88], Mathieu and Domergue defined another generalization of unknotting number. In [Liv02], Livingston worked with this definition.…”

Section: Introductionmentioning

confidence: 99%

Untwisting information from Heegaard Floer homology

Ince

2017

Algebr. Geom. Topol.

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The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. In previous work, we showed that the unknotting and untwisting numbers can be arbitrarily different. In this paper, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to obstruct several 10 and 11-crossing knots from being unknotted by a single positive or negative twist. We also use the Ozsváth-Szabó tau invariant and the Rasmussen s invariant to differentiate between the p-and q-untwisting numbers for certain p, q > 1.

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“…In [MD88], Mathieu and Domergue defined another generalization of unknotting number. In [Liv02], Livingston worked with this definition.…”

Section: Introductionmentioning

confidence: 99%

The untwisting number of a knot

Ince¹

2016

Pacific J. Math.

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Abstract. The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander polynomial-one knot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The untwisting number is the minimum number (over all diagrams of a knot) of rightor left-handed twists on even numbers of strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. We show that the algebraic untwisting number is equal to the algebraic unknotting number. However, we also exhibit several families of knots for which the difference between the unknotting and untwisting numbers is arbitrarily large, even when we only allow twists on a fixed number of strands or fewer.

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Chirurgies de Dehn de pente �1 sur certains n?uds dans lec 3-vari�t�s

Cited by 5 publications

References 7 publications

Slicing knots in definite 4-manifolds

Slicing knots in definite 4-manifolds

Untwisting information from Heegaard Floer homology

The untwisting number of a knot

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