Upsilon-type concordance invariants

Abstract

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.

Connected Floer homology of covering involutions

“…In addition, QA + T is the subgroup generated by alternating knots and torus knots. The following theorem extends results from [1,37]. Theorem 1.6.…”

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

“…Example 4.5. Consider the knot K = 3T (5, 6) # −T (2, 5) # −T (3,5). Applying Theorem 4.3, we have that Υ K (t) = 3Υ T (5,6) (t) − Υ T (3,4) (t) − 3Υ T (2,3) (t).…”

On sums of torus knots concordant to alternating knots

“…Proof of Theorem This is an argument along the line of [, Proposition 1.2]. More precisely, suppose by contradiction that for some , odd, there exists an alternating knot for which the torus knot is concordant to .…”

“…In Section 3 using a connected sum formula for the Kim–Livingston secondary invariant we prove the following theorem, which deals with the first family in Lemma . Theorem For , and odd, is not concordant to a Floer thin knot.…”

“…Consequently, the knots belonging to family (2) and (3) of Lemma (corresponding, respectively, to the cases when is odd and even) cannot be distinguished from a Floer thin knot by means of any upsilon type invariant .…”