Concordance, Crossing Changes, and Knots in Homology Spheres

Davis, Chelsea S.

doi:10.4153/s0008439519000791

Cited by 4 publications

(2 citation statements)

References 18 publications

(42 reference statements)

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“…This conjecture is particularly intriguing because the corresponding statement is known to be false in the smooth category [16,15,25,6]. Evidence for this conjecture was provided in [7,8] and we provide a similar type of evidence in this article. Our evidence will come in the language of Whitney tower concordance.…”

Section: Introductionsupporting

confidence: 72%

The relative Whitney trick and its applications

Davis¹,

Orson²,

Park³

2021

Preprint

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We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a 4-manifold with boundary. Using the relative Whitney trick we prove that every link in a homology sphere is homotopic to a link that is topologically slice in a contractible topological 4-manifold. We further prove that links in homology spheres are equivalent to links in S 3 up to any level of the Whitney tower filtration that approximates link concordance. Finally, we explore the minimum Gordian distance from a link in S 3 to a homotopically trivial link. Extending this notion to links in homology spheres, we use the relative Whitney trick to make explicit computations for 3-component links and establish bounds in general.

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Section: Introductionsupporting

confidence: 72%

The relative Whitney trick and its applications

Davis¹,

Orson²,

Park³

2021

Preprint

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“…Both this and the approach in the introduction are hung up at the homological level. Further discussion of knots in homology spheres and concordance in homology cylinders can be found in, for example, [HLL18], [Dav19].…”

Section: Speculation and Questionsmentioning

confidence: 99%

Deep and shallow slice knots in 4-manifolds

Klug

Benjamin

2021

Proc. Amer. Math. Soc. Ser. B

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We consider slice disks for knots in the boundary of a smooth compact 4-manifold X 4 . We call a knot K ⊂ ∂X deep slice in X if there is a smooth properly embedded 2-disk in X with boundary K, but K is not concordant to the unknot in a collar neighborhood ∂X × I of the boundary.We point out how this concept relates to various well-known conjectures and give some criteria for the nonexistence of such deep slice knots. Then we show, using the Wall self-intersection invariant and a result of Rohlin, that every 4-manifold consisting of just one 0-and a nonzero number of 2-handles always has a deep slice knot in the boundary.We end by considering 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. A generalization of the Murasugi-Tristram inequality is used to show that there does not exist a compact, oriented 4-manifold V with spherical boundary such that every knot K ⊂ S 3 = ∂V is slice in V via a null-homologous disk.

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