Ribbon concordance of fibered knots and compressions of surface homeomorphisms

TL;DR

This paper establishes monotonicity of simplicial volume and dilatation under ribbon concordance of fibered knots, and introduces algorithms for enumerating minimal compressions of surface homeomorphisms, advancing understanding of fibered knots and their concordances.

Contribution

It proves monotonicity properties under ribbon concordance and develops algorithms for minimal compressions, extending previous theorems and providing new tools for knot theory.

Findings

Simplicial volume and dilatation are monotone under ribbon concordance.

Every fibered knot has finitely many predecessors in the ribbon-concordance order.

An algorithm to enumerate minimal compressions of surface homeomorphisms is provided.

Abstract

We prove that simplicial volume and dilatation are monotone under ribbon concordance between fibered knots in $S^3$, and that every fibered knot has only finitely many predecessors in the ribbon-concordance partial order, providing evidence for questions raised by Gordon. We also give an algorithm to enumerate, up to symmetries, all minimal compressions of a surface homeomorphism, extending a theorem of Casson--Long. This yields an algorithm to find all knots that are strongly homotopy-ribbon concordant to a given fibered knot in some homotopy $I\times S^3$. Our study of minimal compressions also provides an alternative perspective on results of Miyazaki concerning nonsimple fibered ribbon knots.