Real Seifert Forms, Hodge Numbers and Blanchfield Pairings

Borodzik, Maciej; Zarzycki, Jakub

doi:10.1007/978-3-030-61958-9_4

Cited by 4 publications

(4 citation statements)

References 21 publications

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“…Note. In [Bor19], Borodzik showed that the minimal number of null-homologous twists on two strands needed to convert a knot K into a knot with Alexander polynomial 1 is always less than three times the algebraic surgery description number. In fact, our Theorem 1.2, together with the fact that a crossing change is a special case of a null-homologous two-strand twist and the fact that u a = tu a , refines this upper bound to twice the algebraic surgery description number.…”

Section: Theorem 27 ([İnc16]mentioning

confidence: 99%

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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: Theorem 27 ([İnc16]mentioning

confidence: 99%

“…The following theorem was inspired by the work of Borodzik on algebraic k-simple knots [Bor19]. In addition, Duncan McCoy suggested the last portion of the proof of Theorem 1.3, improving the upper bound from an earlier version of the paper.…”

Section: An Inequality Relating Surgery Description Number and Untwis...mentioning

confidence: 99%

Unknotting via null-homologous twists and multi-twists

Allen¹,

Ince²,

Kim³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Geometric operations on a knot correspond to algebraic operations on its Seifert matrix, and this in turn leads to the notion of algebraic unknotting. See, for instance, [2,12,21].…”

Section: Introductionmentioning

confidence: 99%

“…This is a topic that has received considerable attention; a few early references include [5,15,22,[24][25][26]. From the four-dimensional perspective, this in turn leads to the question of converting a knot into a knot with Alexander polynomial 1 (which would then be topologically slice [4]); this is explored, for instance, in [2].…”

Section: Introductionmentioning

confidence: 99%