On the Upsilon invariant of fibered knots and right-veering open books

TL;DR

This paper establishes a condition involving the Upsilon invariant that determines when the monodromy of a fibered knot's open book is right-veering, with implications for knot concordance and the Slice-Ribbon Conjecture.

Contribution

It provides a new sufficient condition using the Upsilon invariant for the monodromy to be right-veering, extending previous results on fibered knots and concordance.

Findings

A sufficient condition for right-veering monodromy using Upsilon.

Generalization of Baker's result on ribbon concordances.

Implication that certain fibered knots are unique in their concordance class or challenge the Slice-Ribbon Conjecture.

Abstract

We give a sufficient condition using the Ozsv\'ath-Stipsicz-Szab\'o concordance invariant Upsilon for the monodromy of the open book decomposition of a fibered knot to be right-veering. As an application, we generalize a result of Baker on ribbon concordances between fibered knots. Following Baker, we conclude that either fibered knots $K$ in $S^{3}$ satisfying that $\Upsilon'(t) = -g(K)$ for some $t \in [0,1)$ are unique in their smooth concordance classes or there exists a counterexample to the Slice-Ribbon Conjecture.