Donaldson's theorem, Heegaard Floer homology, and knots with unknotting number one

Greene, Joshua Evan

doi:10.1016/j.aim.2014.01.018

Cited by 15 publications

(29 citation statements)

References 24 publications

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“…Since the boundary Σ(K) is a rational homology 3-sphere, Q K ⊕ R n embeds into the intersection form Q X of X, as can be seen from the Mayer-Vietoris sequence (see [8]). As W K is simply connected (by Theorem 3.2), if X K is simply connected then so too is X.…”

Section: The Case P + Q = −2mentioning

confidence: 99%

“…Unfortunately, however, for this choice of X there always exists an A satisfying (3.1). To get around this problem, we mimic the work of Greene [8], in which Heegaard Floer homology is used to impose a certain structure on A. In order to explain this, we review some Heegaard Floer homology.…”

Section: Plumbingsmentioning

confidence: 99%

“…It is proved in [8]. (Our convention for L(p, q) should be taken as the −p/q surgery on the unknot.)…”

Section: Correction Terms and Sharpnessmentioning

confidence: 99%

“…As mentioned, our first ingredient is the "signed" version of the Montesinos theorem (see Proposition 4.1 of [8] …”

Section: The Case P + Q = −2mentioning

confidence: 99%

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Pretzel knots with unknotting number one

Buck

Gibbons

Staron

2013

Communications in Analysis and Geometry

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We provide a partial classification of the 3-strand pretzel knots K = P (p, q, r) with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with r even. We discover that there are only four possible subfamilies which may satisfy u(K) = 1. These families are determined by the sum p + q and their signature, and we resolve the problem in two of these cases. Ingredients in our proofs include Donaldson's diagonalization theorem (as applied by Greene), Nakanishi's unknotting bounds from the Alexander module, and the correction terms introduced by Ozsváth and Szabó. Based on our results and the fact that the 2-bridge knots with unknotting number one are already classified, we conjecture that the only 3-strand pretzel knots P (p, q, r) with unknotting number one that are not 2-bridge knots are P (3, −3, 2) and its reflection.

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Section: The Case P + Q = −2mentioning

confidence: 99%

Section: Plumbingsmentioning

confidence: 99%

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Pretzel knots with unknotting number one

Buck

Gibbons

Staron

2013

Communications in Analysis and Geometry

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“…This is the obvious generalisation to links of the unknotting number, and one should expect that many of the same methods should apply to compute it for examples. There have been several successful applications of Donaldson's diagonalisation theorem and Heegaard Floer homology to the calculation of unknotting numbers, for example [6,26,21,11,19]. It is an interesting problem, which we begin to address here, to generalise these techniques to the case of links.…”

Section: Introductionmentioning

confidence: 99%

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Nagel

Owens²

2015

Bulletin of the London Mathematical Society

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Alternating knots with unknotting number one

McCoy

2017

Advances in Mathematics

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We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alternating knots. As a consequence, we also get a converse to the Montesinos trick: an alternating knot has unknotting number one if its branched double cover arises as half-integer surgery on a knot in S 3 . We also reprove a characterisation of almost-alternating diagrams of the unknot originally due to Tsukamoto.

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Donaldson's theorem, Heegaard Floer homology, and knots with unknotting number one

Cited by 15 publications

References 24 publications

Pretzel knots with unknotting number one

Pretzel knots with unknotting number one

Unlinking information from 4-manifolds

Alternating knots with unknotting number one

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