On sums of torus knots concordant to alternating knots

Aceto, Paolo; Alfieri, Antonio

doi:10.1112/blms.12228

Cited by 4 publications

(3 citation statements)

References 33 publications

(75 reference statements)

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“…The connected sum formula (Theorem 6.2) employed in the proof of Proposition 1.2 is used in [1] to decide which sums of two torus are concordant to alternating knots.…”

Section: Introductionmentioning

confidence: 99%

Upsilon-type concordance invariants

Alfieri

2019

Algebr. Geom. Topol.

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To a region C of the plane satisfying a suitable convexity condition we associate a knot concordance invariant Υ C . For appropriate choices of the domain this construction gives back some known knot Floer concordance invariants like Rasmussen's h i invariants, and the Ozsváth-Stipsicz-Szabó upsilon invariant. Furthermore, to three such regions C, C + and C − we associate invariants Υ C ± ,C generalising Kim-Livingston secondary invariant. We show how to compute these invariants for some interesting classes of knots (including alternating and torus knots), and we use them to obstruct concordances to Floer thin knots and algebraic knots.1.1. In [8] Lidman and Moore characterized L-space pretzel knots. They found that a pretzel knot has an L-space surgery if and only if it is a torus knot T 2,2n+1 for some n ≥ 1, or a pretzel knot in the form P (−2, 3, q) for some q ≥ 7 odd. Motivated by the exploration started by Wang [27], and Livingston [10] one may wonder if L-space pretzel knots of the form P (−2, 3, q) are concordant to algebraic knots. Theorem 1.1. None of the L-space pretzel knots P (−2, 3, q), with q ≥ 7 odd, is conconcordant to a sum of algebraic knots.Notice that for these knots the obstruction found in [27, Corollary 3.5] vanish. In [11]Friedl, Livingston and Zentner asked whenever a sum of torus knots is concordant to an alternating knot. In [28] Zemke used involutive Floer homology [5] to prove that certain connected sums of torus knots are not concordant to Floer thin knots. Floer thin knots are upsilon-alternating, meaning that Υ K (t) = −τ (K) · (1 − |1 − t|). A straightforward argument shows that a sum of positive torus knots is upsilon-alternating if and only if it is a connected sum of (2, 2n + 1) torus knots and indeed alternating. However, when both positive and negative torus knots are involved this obstruction can vanish.Proposition 1.2. The knot K = T 8,5 # − T 6,5 # − T 4,3 is upsilon-alternating but not concordant to a Floer thin knot.

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“…The connected sum formula (Theorem 6.2) employed in the proof of Proposition 1.2 is used in [1] to decide which sums of two torus are concordant to alternating knots.…”

Section: Introductionmentioning

confidence: 99%

Upsilon-type concordance invariants

Alfieri

2019

Algebr. Geom. Topol.

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“…By [AA19, Corollary 1.5], we only need to show that is not concordant to a connected sum of -bridge knots for any . If it is, then by Corollary 1.7 there exists a connected sum of -bridge knots concordant to and such that divides .…”

Section: Consequences Of Theorem 11mentioning

confidence: 99%

Rational cobordisms and integral homology

Aceto¹,

Celoria²,

Park³

2020

Compositio Math.

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We consider the question of when a rational homology $3$ -sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$ -bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

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“…For instance, Friedl-Livingston-Zentner [7] show that alternating knots generate a subgroup with infinitely generated quotient in the concordance group. Aceto-Alfieri [1] have studied the question (given in [7]) of which sums of torus knots are concordant to alternating knots. Similar questions are addressed in, for instance, [2,3,4,21].…”

Section: Introductionmentioning

confidence: 99%

Concordances from differences of torus knots to $L$-space knots

Allen

2019

Proc. Amer. Math. Soc.

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It is known that connected sums of positive torus knots are not concordant to L-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial L-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.

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On sums of torus knots concordant to alternating knots

Cited by 4 publications

References 33 publications

Upsilon-type concordance invariants

Upsilon-type concordance invariants

Rational cobordisms and integral homology

Concordances from differences of torus knots to $L$-space knots

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