On concordances in 3-manifolds

Celoria, Daniele

doi:10.1112/topo.12051

Cited by 12 publications

(10 citation statements)

References 44 publications

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“…We observe that almost-concordant knots lift to links for which the respective sets of pairwise linking numbers coincide, and the result follows. We remark that the examples exhibited by Celoria [Cel16] also lift to links in S 3 with different pairwise linking numbers. So in particular these examples are distinct in topological almost-concordance, a fact which cannot be detected by the τ invariant.…”

Section: Introductionmentioning

confidence: 70%

“…Celoria [Cel16], who coined the term 'almost-concordance', recently studied the case of the null-homotopic class in lens spaces L(n, 1) with n ≥ 3. Using a generalisation of the τ invariant from knot Floer homology, he showed the existence of an infinite family of knots, mutually distinct in smooth almost-concordance.…”

Section: Introductionmentioning

confidence: 99%

“…Using a generalisation of the τ invariant from knot Floer homology, he showed the existence of an infinite family of knots, mutually distinct in smooth almost-concordance. In [Cel16,Conjecture 44] it is conjectured that, when Y = S 3 , within each free homotopy class in [S 1 , Y ] there are infinitely many distinct almost-concordance classes. For Y = S 3 the local action is transitive.…”

Section: Introductionmentioning

confidence: 99%

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Satellites and concordance of knots in 3–manifolds

Friedl

Nagel

Orson

et al. 2018

Trans. Amer. Math. Soc.

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Given a 3-manifold Y and a free homotopy class in [S 1 , Y ], we investigate the set of topological concordance classes of knots in Y × [0, 1] representing the given homotopy class. The concordance group of knots in the 3-sphere acts on this set. We show in many cases that the action is not transitive, using two techniques. Our first technique uses Reidemeister torsion invariants, and the second uses linking numbers in covering spaces. In particular, we show using covering links that for the trivial homotopy class, and for any 3-manifold that is not the 3-sphere, the set of orbits is infinite. On the other hand, for the case that Y = S 1 × S 2 , we apply topological surgery theory to show that all knots with winding number one are concordant.

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Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Satellites and concordance of knots in 3–manifolds

Friedl

Nagel

Orson

et al. 2018

Trans. Amer. Math. Soc.

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“…Finally, it is clear that there is a 0-surgery on an l-component sublink of L i gives the sum K#K ⊂ # S 2 × S 1 and thus applying Theorem 4.7 again we have D(K) = D(K#K ) when K is a local knot. Therefore Theorem 4.8 also gives us a knot in # i S 2 × S 1 for each i which not almost-concordant in # i S 2 × S 1 to the unknot in the sense of [Cel18].…”

Section: A Generalization Of the First Non-vanishing Milnor's Invariantmentioning

confidence: 92%

An Analogue of Milnor's Invariants for Knots in 3-Manifolds

Kuzbary¹

2019

Preprint

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Milnor's invariants are some of the more fundamental oriented link concordance invariants; they behave as higher order linking numbers and can be computed using combinatorial group theory (due to Milnor), Massey products (due to Turaev and Porter), and higher order intersections (due to Cochran). In this paper, we generalize the first non-vanishing Milnor's invariants to oriented knots inside a closed, oriented 3-manifold M . We call this the Dwyer number of a knot and show methods to compute it for null-homologous knots inside connected sums of S 2 × S 1 . We further show in this case the Dwyer number provides the weight of the first non-vanishing Massey product in the knot complement in the ambient manifold. Additionally, we prove the Dwyer number detects a family of knots K in # S 2 × S 1 bounding smoothly embedded disks in S 2 × D 2 which are not concordant to the unknot.

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“…Given K ⊂ S 2 × S 1 , the connected sum with a local knot K ′ ⊂ S 3 does not alter the geometric winding number. Hence any concordance bound obtained on ⋔(K) is in fact an almostconcordance bound, using the terminology of [5].…”

Section: Preliminariesmentioning

confidence: 98%

Heegaard Floer homology and concordance bounds on the Thurston norm

Celoria

Golla

2019

Trans. Amer. Math. Soc.

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We prove that twisted correction terms in Heegaard Floer homology provide lower bounds on the Thurston norm of certain cohomology classes determined by the strong concordance class of a 2-component link L in S 3 . We then specialise this procedure to knots in S 2 × S 1 , and obtain a lower bound on their geometric winding number. Furthermore we produce an obstruction for a knot in S 3 to have untwisting number 1. We then provide an infinite family of nullhomologous knots with increasing geometric winding number, on which the bound is sharp.

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On concordances in 3-manifolds

Cited by 12 publications

References 44 publications

Satellites and concordance of knots in 3–manifolds

Satellites and concordance of knots in 3–manifolds

An Analogue of Milnor's Invariants for Knots in 3-Manifolds

Heegaard Floer homology and concordance bounds on the Thurston norm

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