Secondary Upsilon invariants of knots

Kim, Se-Goo; Livingston, Charles

doi:10.1093/qmath/hax062

Cited by 15 publications

(30 citation statements)

References 17 publications

(46 reference statements)

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“…Summarizing, given south-west regions C + , C − and C ⊂ R 2 we get a map Υ C ± ,C : CFK/ ∼ → [−∞, +∞). In [7] Kim and Livingston produce south-west regions for which the condition Z + ∩ Z − = ∅ is guaranteed.…”

Section: Secondary Invariantsmentioning

confidence: 99%

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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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Section: Secondary Invariantsmentioning

confidence: 99%

“…See [23] for an extensive exposition of this topic. Knot Floer homology has been used to produce knot concordance invariants by many authors [22,15,24,7]. The purpose of this note is to show that all these constructions can be seen as particular cases of a more general construction.…”

Section: Introductionmentioning

confidence: 99%

Upsilon-type concordance invariants

Alfieri

2019

Algebr. Geom. Topol.

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“…We compute that L 4 5 ,s has slope m = − 3 2 and j-intercept b = 5s 2 . In Figure 4, one can see that L 4 5 ,s with minimal s passes through the points (1,8) and (3,5). The j-intercept of this line is 19 2 corresponding to an s value of 19 5 .…”

Section: Resultsmentioning

confidence: 94%

“…Recently, Feller and Krcatovich (in [4]) determined relationships among the Upsilon functions of torus knots. Our goal here is to use the secondary Upsilon invariants, defined by Kim and Livingston in [8], to show that these relationships do not extend to stabilized knot complexes of torus knots.…”

Section: Introductionmentioning

confidence: 99%

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Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

Allen

2020

Algebr. Geom. Topol.

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Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants τ , ε, and Υ. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants defined by Kim and Livingston to show that these relations do not extend to stable equivalence.

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The $$\nu ^+$$-equivalence classes of genus one knots

Sato

2023

Sel. Math. New Ser.

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Secondary Upsilon invariants of knots

Cited by 15 publications

References 17 publications

Upsilon-type concordance invariants

Upsilon-type concordance invariants

Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

The $$\nu ^+$$-equivalence classes of genus one knots

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