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The signature and cusp geometry of hyperbolic knots

Alex Davies, András Juhász, Marc Lackenby, Nenad Tomasev

TL;DR

The paper defines the natural slope, a cusp-geometric invariant of hyperbolic knots, and proves that the discrepancy between twice the knot signature and the natural slope is controlled by hyperbolic volume and injectivity radius via |$2\sigma(K)$ - $\operatorname{slope}(K)$| ≤ $c_1\operatorname{vol}(K)\operatorname{inj}(K)^{-3}$. It then refines this with a slope-correction using short geodesics (the OddGeo term) to obtain a bound independent of the injectivity radius, and derives applications to Dehn surgery and the 4-ball genus. A key part of the approach combines geometric triangulations and Gordon–Litherland signatures, enabling a linear-in-volume bound on the difference between slope and a signature-based quantity. The work is supported by machine-learning discovery that initially linked $\operatorname{Re}(\mu)$ to $2\sigma(K)$, and it extends to highly twisted knots, spanning surfaces, and experimental data on random knots, highlighting deep connections between hyperbolic geometry and 4-dimensional knot invariants with practical implications for Dehn surgery and concordance theory.

Abstract

We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We also show a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times.

The signature and cusp geometry of hyperbolic knots

TL;DR

The paper defines the natural slope, a cusp-geometric invariant of hyperbolic knots, and proves that the discrepancy between twice the knot signature and the natural slope is controlled by hyperbolic volume and injectivity radius via | - | ≤ . It then refines this with a slope-correction using short geodesics (the OddGeo term) to obtain a bound independent of the injectivity radius, and derives applications to Dehn surgery and the 4-ball genus. A key part of the approach combines geometric triangulations and Gordon–Litherland signatures, enabling a linear-in-volume bound on the difference between slope and a signature-based quantity. The work is supported by machine-learning discovery that initially linked to , and it extends to highly twisted knots, spanning surfaces, and experimental data on random knots, highlighting deep connections between hyperbolic geometry and 4-dimensional knot invariants with practical implications for Dehn surgery and concordance theory.

Abstract

We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. We show that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We also show a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times.
Paper Structure (12 sections, 18 theorems, 91 equations, 13 figures)

This paper contains 12 sections, 18 theorems, 91 equations, 13 figures.

Table of Contents

  1. Introduction
  2. An application to Dehn surgery
  3. An application to 4-ball genus
  4. Spanning surfaces
  5. Highly twisted knots
  6. Methodology
  7. Hyperbolic knots and natural slope
  8. Proof of Theorem \ref{['thm:crosscap']}
  9. The knot signature
  10. Highly twisted knots
  11. Proof of Theorem \ref{['Thm:SignatureCorrection']}
  12. Experimental data and some conjectures about random knots

Key Result

Theorem 1.1

There exists a constant $c_1$ such that, for any hyperbolic knot $K$,

Figures (13)

  • Figure 1: A geodesic running in the direction $\mu^\perp$ that is perpendicular to the meridian $\mu$. By the time it returns to the meridian, it has travelled one longitude minus some multiple $s$ of the meridian. This real number $s$ is the natural slope of $K$.
  • Figure 2: A plot of signature versus the real part of the meridional translation, $\operatorname{Re}(\mu)$, coloured by longitudinal translation, for a dataset of knots randomly generated by SnapPy.
  • Figure 3: A plot of signature versus slope for knots up to 16 crossings in the Regina census (left) and for a dataset of knots randomly generated by SnapPy having 10 to 80 crossings in their SnapPy-simplified form (right).
  • Figure 4: Left: The stevedore knot $6_1$, which is a slice knot. Right: Its cusp torus, as provided by SnapPy SnapPy. The longitude is $3.9279$ and the meridian is $0.7237 + 1.0160i$. Its natural slope is $1.8267$ and its signature is $0$.
  • Figure 5: Left: The knot 12a52. Right: Its cusp torus. The longitude is $27.7228$ and the meridian is $-1.2838 + 0.5145i$. Its natural slope is $-18.6064$ and its signature is $-8$. Note how far the parallelogram is from being right-angled; this is the defining feature of having very positive or very negative slope.
  • ...and 8 more figures

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.3
  • ...and 30 more