Ribbon concordance and fibered predecessors, II: the general case

TL;DR

This paper proves that any knot in S^3 has only finitely many fibered predecessors under ribbon concordance, extending previous results by removing the hyperbolic restriction and introducing new Floer homology inequalities.

Contribution

It establishes a finiteness result for fibered predecessors of knots without hyperbolic constraints, using novel inequalities in knot Floer homology via bordered Heegaard Floer techniques.

Findings

Finiteness of fibered predecessors for all knots in S^3.

An inequality relating knot Floer homology of satellite knots and their companions.

Explicit upper bounds on the Gromov norm of knot complements.

Abstract

The first and third authors recently proved that for each knot $K\subset S^3$ there are only finitely many hyperbolic fibered knots which are ribbon concordant to $K$. In this paper, we remove the hyperbolic constraint, proving that every knot in $S^3$ has only finitely many fibered predecessors under ribbon concordance. The key new input is an inequality relating the knot Floer homology of a generalized satellite knot with that of its companion, proved via the immersed curves formulation of bordered Heegaard Floer homology, which should be of independent interest. Our work, together with results of Kojima--McShane, also leads to an explicit upper bound on the Gromov norm of the complement of any fibered predecessor of a knot $K \subset S^3$, in terms of the arc index and genus of $K$.