Nonsurjective Satellite Operators and Piecewise-Linear Concordance

Levine, Adam Simon

doi:10.1017/fms.2016.31

Cited by 34 publications

(42 citation statements)

References 36 publications

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“…on the set of knot concordance classes C. If r = 0, then the subscript is often suppressed. Some of our results are especially interesting in light of recent research on the injectivity and surjectivity of such satellite operators [9,11,12,25]. For example, reinterpreting Corollary 3.10 in this language for r = 0 yields the following.…”

Section: Introductionmentioning

confidence: 78%

Shake slice and shake concordant knots

Cochran

Ray

2016

J Topology

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A crucial step in the surgery-theoretic program to classify smooth manifolds is that of representing a middle-dimensional homology class by a smoothly embedded sphere. This step fails even for the simple 4-manifolds obtained from the 4-ball by adding a 2-handle with framing r along some knot K → ∂B 4 . An r-shake slice knot is one for which a generator of the second homology of this 4-manifold can be represented by a smoothly embedded 2-sphere. It is not known whether there exist 0-shake slice knots that are not slice. We define a relative notion of r-shake sliceness of knots, which we call r-shake concordance which is easily seen to be a generalization of classical concordance, and we give the first examples of knots that are 0-shake concordant but not concordant; these may be chosen to be topologically slice. Additionally, for each r we completely characterize r-shake slice and r-shake concordant knots in terms of concordance and satellite operators. Our characterization allows us to construct new families of possible r-shake slice knots that are not slice. Rice University). K denote the 3-manifold obtained by performing r-framed surgery on S 3 along the knot K.Recall that the classical concordance invariants are determined by the homology cobordism type of M 0 K , along with the homology class of the positive meridian. Hence the question of whether there exist 0-shake slice knots that are not slice is especially difficult because there are presently no known invariants that can distinguish between a knot being slice in B 4 and it being slice in merely a homology B 4 , except (possibly) Rasmussen's s -invariant and its generalizations.Therefore, it may be surprising that here we find success in the relative case.Theorem 4.1. For any integer r, there exist infinitely many knots which are distinct in smooth concordance but are pairwise r-shake concordant. For r = 0, there exist topologically slice knots with this property as well. In addition, for any integer r, none of τ, s , or slice genus is invariant under r-shake concordance.This result can be seen as a consequence of our complete characterization of r-shake concordance in terms of concordance and certain winding number one satellites. Let P be a pattern knot, that is, a knot inside a solid torus (an example is shown in Figure 1). For any knot K, let P r (K) denote the r-twisted satellite of K with pattern P [27, p. 10, 37, p. 110]. The symbol P (K) denotes the 0-twisted (that is, classical) satellite of K with pattern P . Let P denote the knot in S 3 given by P when the solid torus is placed in S 3 in the standard unknotted manner, or equivalently, P = P (U ), where U is the unknot.Theorem 3.7. For any integer r, the knots K and J are r-shake concordant if and only if there exist winding number one patterns P and Q, with P and Q ribbon knots, such that P r (K) is concordant to Q r (J).Corollary 3.9. For any integer r, the equivalence relation on the set of knots generated by r-shake concordance is the same as that generated by concordance together with the ...

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Section: Introductionmentioning

confidence: 78%

Shake slice and shake concordant knots

Cochran

Ray

2016

J Topology

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“…In , Levine proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in

S^{3}

, even if one allows concordances in smooth homology cobordisms. Doing so answered a question of Matsumoto [, Problem 1.31].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…It follows immediately from the definition of

{\hat{scriptC}}_{sm}

that for every knot

K

M

(M, K)

is not smoothly homology concordant to any knot in

S^{3}

meaning

(M, K)

is not in the image of

Ψ_{sm}

. Even more strongly, by work of Levine [, Theorem 1.1], there exists a pair

false(M, K false) \in {\overset{C}{true}}_{sm}

for which

M

is smoothly homology cobordant to

S^{3}

and yet

(M, K)

does not cobound a smooth annulus with any knot in

S^{3}

in any smooth homology cobordism.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…If

normalΨ

were surjective, then the same diagram chase would give that

P : {scriptC}_{double-struckZ} \to {scriptC}_{double-struckZ}

is bijective. This would be noteworthy because as a consequence of there exists a winding number 1 pattern

P

for which

K \mapsto P (K)

is not surjective on smooth integral knot concordance. In Section , we recall the notions of , verify that these notions are compatible with the solvable filtration, and use that

Ψ : C / {scriptF}_{n} \to \hat{scriptC} / {\overset{F}{true}}_{n}

is a bijection by Theorem to prove the following.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Winding number ±1 patterns are more subtle. In [12], a winding number 1 pattern is produced which does not give a surjection on C sm . This section can be thought of as evidence that no such pattern exists for topological concordance, so that every P : C → C is a bijection.…”

Section: Application: Bijective Satellite Operatorsmentioning

confidence: 99%

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Topological concordance of knots in homology spheres and the solvable filtration

Davis

2020

Journal of Topology

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In 2016, Levine showed that there exists a knot in a homology 3‐sphere which is not smoothly concordant to any knot in S3 where one allows concordances in any smooth homology cobordism. Whether the same is true if one allows topological concordances is not known. One might hope that such an example might be detected by the powerful filtration of knot concordance introduced by Cochran–Orr–Teichner. We prove that this is not the case, demonstrating that for any knot in any homology sphere there is a knot in S3 equivalent to the original knot modulo any term of this filtration. Our results apply equally well to link concordance. As an application, we prove that every winding number ±1 satellite operator acts bijectively on knot concordance, modulo any term of the solvable filtration.

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Knot Floer homology of satellite knots with (1, 1) patterns

Chen

2023

Sel. Math. New Ser.

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Nonsurjective Satellite Operators and Piecewise-Linear Concordance

Cited by 34 publications

References 36 publications

Shake slice and shake concordant knots

Shake slice and shake concordant knots

Topological concordance of knots in homology spheres and the solvable filtration

Knot Floer homology of satellite knots with (1, 1) patterns

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