On cobordisms between knots, braid index, and the Upsilon-invariant

Abstract

We use Ozsváth, Stipsicz, and Szabó's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a positive full-twist realize the braid index of their closure. We also establish that quasi-positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure. Finally, we provide inductive formulas for the Upsilon invariant of torus knots and compare it to the Levine-Tristram signature profile.

25 pages, 3 figures, comments welcome! Second version: Typos fixed, implementation of recommendations. Accepted for publication by Mathematische Annalen

References in corpus (1)

• Algebraic functions and closed braids

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