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Ribbon concordance and fibered predecessors, II: the general case

John A. Baldwin, Jonathan Hanselman, Steven Sivek

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 $K$.

Ribbon concordance and fibered predecessors, II: the general case

Abstract

The first and third authors recently proved that for each knot there are only finitely many hyperbolic fibered knots which are ribbon concordant to . In this paper, we remove the hyperbolic constraint, proving that every knot in 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 , in terms of the arc index and genus of .
Paper Structure (16 sections, 28 theorems, 206 equations, 3 figures)

This paper contains 16 sections, 28 theorems, 206 equations, 3 figures.

Table of Contents

  1. Introduction
  2. On the proof
  3. A technical result and a question
  4. Concluding remarks
  5. Conventions
  6. Organization
  7. Acknowledgements
  8. Branched cyclic covers and knot Floer homology
  9. Generalized satellite knots
  10. Fibered satellite knots and reducible monodromy
  11. Immersed curves and a satellite inequality
  12. Bordered Heegaard Floer homology
  13. An immersed curves formulation
  14. Restrictions on immersed curves
  15. Proof of Theorem \ref{['thm:gensat']}
  16. ...and 1 more sections

Key Result

Theorem 1.2

For each knot $K\subset S^3$, there are only finitely many fibered knots $J\leq K$.

Figures (3)

  • Figure 1: An example of a Nielsen--Thurston decomposition.
  • Figure 2: Local pictures of the moves we use to simplify the multicurve $\Gamma$.
  • Figure 3: Taking a pair of curves $\Gamma_P$ and $\Gamma_C$, shown at the top in the universal cover of $T_{\bullet}$, and then applying a sequence of the moves (i)-(v) in order to turn $\Gamma_P$ into $\Gamma_\mu$ without ever increasing the intersection number with $\Gamma_C$.

Theorems & Definitions (61)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • ...and 51 more