A further note on the concordance invariants epsilon and upsilon

Wang, Shida

doi:10.1090/proc/14727

Cited by 4 publications

(1 citation statement)

References 19 publications

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“…Thus two hyperbolic L-space knots in our pair are not concordant. In the literature, there are plenty of examples of non-concordant knots sharing the same Upsilon invariant [1,11,17,38,39,40,42]. However, they use either connected sums of torus knots or satellite knots, which are not hyperbolic.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic L-space knots and their formal semigroups

Teragaito

2022

Int. J. Math.

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For an L-space knot, the formal semigroup is defined from its Alexander polynomial. It is not necessarily a semigroup. That is, it may not be closed under addition. There exists an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by three elements. In this paper, we give the first infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by five elements.

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Section: Introductionmentioning

confidence: 99%

Hyperbolic L-space knots and their formal semigroups

Teragaito

2022

Int. J. Math.

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An infinite family of chiral, non-slice, non-alternating hyperbolic knots with vanishing Upsilon invariants

Teragaito

2024

Bol. Soc. Mat. Mex.

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For a knot in the 3-sphere, its Upsilon invariant is a continuous, piecewise linear function defined on the interval [0,2]. It is a concordance invariant, so vanishes for slice knots. Since it only changes the sign for the mirror image with reversed orientation, it also vanishes for amphicheiral knots. For alternating knots, more generally, quasi-alternating knots, the Upsilon invariant is determined only by the signature. In particular, it vanishes if such a knot has zero signature. On the other hand, it is known that the Upsilon invariant is a non-zero convex function for L-space knots. In this paper, we construct an infinite family of hyperbolic knots, each of which has zero Upsilon invariant, but is chiral, non-slice, non-alternating. To confirm that the Upsilon invariant vanishes, we calculate the full knot Floer complex.

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A Note on the Concordance Invariant Epsilon

Wang

2021

The Quarterly Journal of Mathematics

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We compare the smooth concordance invariants Upsilon, phi and epsilon. Previous work gave examples of knots with one of the Upsilon and phi invariants zero but the epsilon invariant nonzero. We build an infinite family of linearly independent knots with both the Upsilon and phi invariants zero but the epsilon invariant nonzero. This provides examples of knots with arbitrarily large concordance genus but vanishing bounds from the Upsilon and phi invariants.

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A further note on the concordance invariants epsilon and upsilon

Abstract: Hom gives an example of a knot with vanishing Upsilon invariant but nonzero epsilon invariant. We build more such knots that are linearly independent in the smooth concordance group.

Cited by 4 publications

References 19 publications

Hyperbolic L-space knots and their formal semigroups

Hyperbolic L-space knots and their formal semigroups

An infinite family of chiral, non-slice, non-alternating hyperbolic knots with vanishing Upsilon invariants

A Note on the Concordance Invariant Epsilon

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