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Non-integer characterizing slopes and knot Floer homology

Duncan McCoy

TL;DR

This work advances the understanding of characterizing slopes for knots in S^3 by marrying JSJ decomposition techniques with knot Floer homology. It proves that almost all non-integer slopes are characterizing for any knot with a simple Floer-theoretic profile, and identifies broad classes (alternating, L-space, almost L-space, and many small-crossing primes) for which this holds; it also establishes infinitely many integer characterizing slopes for all L-space and almost L-space knots. The analysis hinges on Property SpliF F, a structural condition on knot Floer data, and on careful control of Dehn surgery via the outermost JSJ piece, V_k sequences, and d-invariants. The results provide explicit bounds and techniques that extend known cases (torus, hyperbolic L-space, composites) to satellites and more general knots, contributing a substantial step toward the conjecture that non-integer slopes are characterizing for essentially all knots. Overall, the paper offers concrete, computable criteria for when a knot and its mirror yield unique surgery manifolds at a given slope, with significant implications for understanding Dehn surgery and knot invariants through Heegaard Floer theory.

Abstract

Conjecturally, a knot in the 3-sphere has only finitely many non-integer non-characterizing slopes. We verify this conjecture for all knots with knot Floer homology satisfying certain simplicity conditions. The class of knots satisfying our notion of simplicity includes alternating knots, $L$-space knots and the vast majority of knots with at most 12 crossings. For arbitrary knots in the 3-sphere we show that almost all slopes $p/q$ with $|q|\geq 3$ are characterizing. In addition, we show that all $L$-space knots and almost $L$-space knots have infinitely many integer characterizing slopes.

Non-integer characterizing slopes and knot Floer homology

TL;DR

This work advances the understanding of characterizing slopes for knots in S^3 by marrying JSJ decomposition techniques with knot Floer homology. It proves that almost all non-integer slopes are characterizing for any knot with a simple Floer-theoretic profile, and identifies broad classes (alternating, L-space, almost L-space, and many small-crossing primes) for which this holds; it also establishes infinitely many integer characterizing slopes for all L-space and almost L-space knots. The analysis hinges on Property SpliF F, a structural condition on knot Floer data, and on careful control of Dehn surgery via the outermost JSJ piece, V_k sequences, and d-invariants. The results provide explicit bounds and techniques that extend known cases (torus, hyperbolic L-space, composites) to satellites and more general knots, contributing a substantial step toward the conjecture that non-integer slopes are characterizing for essentially all knots. Overall, the paper offers concrete, computable criteria for when a knot and its mirror yield unique surgery manifolds at a given slope, with significant implications for understanding Dehn surgery and knot invariants through Heegaard Floer theory.

Abstract

Conjecturally, a knot in the 3-sphere has only finitely many non-integer non-characterizing slopes. We verify this conjecture for all knots with knot Floer homology satisfying certain simplicity conditions. The class of knots satisfying our notion of simplicity includes alternating knots, -space knots and the vast majority of knots with at most 12 crossings. For arbitrary knots in the 3-sphere we show that almost all slopes with are characterizing. In addition, we show that all -space knots and almost -space knots have infinitely many integer characterizing slopes.
Paper Structure (14 sections, 33 theorems, 107 equations, 1 table)

This paper contains 14 sections, 33 theorems, 107 equations, 1 table.

Table of Contents

  1. Introduction
  2. Property Spli$\mathbb{F}$F
  3. Integer characterizing slopes
  4. Proof of Theorems \ref{['thm:g_geq_3']} and \ref{['thm:spaff_slopes']}
  5. Surgery on knots of bounded genus
  6. Knots of hyperbolic type
  7. Surgery and the JSJ decomposition
  8. Proof of Theorem \ref{['thm:bounded_genus']}
  9. Knot Floer homology and Property Spli$\mathbb{F}$F
  10. Knot Floer homology
  11. The mapping cone formula
  12. Thickness one knots.
  13. Applications to characterizing slopes
  14. Integral slopes on almost $L$-space knots

Key Result

Theorem 1.2

Let $K$ be a knot in $S^3$. Then there exists a constant $C=C(K)$ such that any slope $p/q$ satisfying $|q|\geq 3$ and $|p|+|q|\geq C$ is characterizing for $K$.

Theorems & Definitions (74)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.3
  • Definition 1.4
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6: Sorya, Sorya2023satellite
  • Theorem 1.7: Sorya, Sorya2023satellite
  • Remark 1.8
  • ...and 64 more