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

Peter Feller,David Krcatovich

doi:10.1007/s00208-017-1519-1

Peter Feller, David Krcatovich

Open Access

https://doi.org/10.1007/s00208-017-1519-1

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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.

Highlights

This article is concerned with the study of knots in the 3-sphere S3—smooth oriented embeddings of the circle S1 considered up to ambient isotopy
We reprove and generalize a classical consequence of the Morton-FranksWilliams inequality [Mor86, FW87] and we find that most torus knots minimize the braid index among all knots concordant to them; see Theorem 1.3
We use Ozsvath, Stipsicz, and Szabo’s Υ-invariant to improve the triangle inequality (1) by a term depending on the braid indices when the involved knots are quasi-positive—knots K for which there exist positive integers n and l such that K is the braid closure β of an n-braid β given as the product of l conjugates of the standard generators ai of the braid group on n strands

Summary

Introduction

This article is concerned with the study of knots in the 3-sphere S3—smooth oriented embeddings of the circle S1 considered up to ambient isotopy. |g4(K) − g4(T )| ≤ d(K, T ) = g4(K m(T )) ≤ g4(K) + g4(T ) In this text, we use Ozsvath, Stipsicz, and Szabo’s Υ-invariant to improve the triangle inequality (1) by a term depending on the braid indices when the involved knots are quasi-positive—knots K for which there exist positive integers n and l such that K is the braid closure β of an n-braid β given as the product of l conjugates of the standard generators ai of the braid group on n strands. We show that sufficiently twisted quasi-positive n-braid closures cannot be concordant to any knot of smaller braid index: Theorem 1.3. If yes, this would imply that Corollary 1.2 holds true without any assumption on K, and, in particular, that each torus knot realizes the minimal braid index among all knots in its concordance class.

Υ for torus knots

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