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

TL;DR

This paper explores how the Upsilon-invariant can bound cobordisms between knots with full-twists, generalize classical inequalities, and analyze braid index and concordance properties of certain braids, with explicit formulas for torus knots.

Contribution

It introduces new bounds on knot cobordisms using the Upsilon-invariant, generalizes classical braid index results, and provides inductive formulas for the Upsilon-invariant of torus knots.

Findings

Positive braids with full-twists realize their braid index.

Sufficiently twisted quasi-positive braids achieve minimal braid index among concordant knots.

Derived inductive formulas for the Upsilon-invariant of torus knots.

Abstract

We use Ozsv\'ath, Stipsicz, and Szab\'o'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.