Ribbon concordance and fibered predecessors

TL;DR

The paper addresses whether a knot $K$ in $S^3$ has only finitely many hyperbolic fibered predecessors under ribbon concordance. It introduces a method combining the injectivity of knot Floer invariants under ribbon cobordisms with a link between the knot Floer homology of a fibered knot and fixed points of its monodromy, yielding a bound $\lambda(J) \le \delta!$ where $\delta$ is the arc index of $K$, and hence finitely many possible monodromies up to conjugacy. By leveraging entropy-volume relations from Kojima–McShane and Cornish, the work translates dilatation bounds into volume constraints for hyperbolic predecessors and establishes finiteness results for fibered knots. The Discussion then outlines extensions to nonfibered cases and raises questions about monotonicity of simplicial volume and the guts under ribbon concordance, highlighting directions for future work.

Abstract

Given any knot K in the 3-sphere, we prove that there are only finitely many hyperbolic fibered knots which are ribbon concordant to K. It follows that every fibered knot in the 3-sphere has only finitely many hyperbolic predecessors under ribbon concordance. Our proof combines results about maps on Floer homology induced by ribbon cobordisms with a relationship between the knot Floer homology of a fibered knot and fixed points of its monodromy. We then use the same techniques in combination with results of Cornish and Kojima-McShane to prove an inequality relating the volumes of ribbon concordant hyperbolic fibered knots.

Theorems & Definitions (24)

• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Lemma 2.1
• proof
• Lemma 2.2
• proof