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Euclidean lengths and the Culler-Shalen norms of slopes

Kazuhiro Ichihara

TL;DR

This work investigates two slope-valued invariants in hyperbolic knot complements: the Euclidean length on a horotorus, length_T(r), and the Culler–Shalen norm, ||r||. It proves inequalities linking these quantities, notably a lower bound ||r|| ≥ (2/3) length_T(r) for certain knot exteriors and general two-slope configurations, and shows how length and norm bounds constrain boundary slopes and the boundary-slope diameter. The paper also derives general length–norm comparison results for numerical slopes, including equality cases under extremal slope conditions, and applies these to bound Diam(B_M). The figure-eight knot exterior serves as a concrete example demonstrating the sharpness and limitations of the bounds, with explicit formulas for length_T and ||r|| and a precise Diam(B_M) value.

Abstract

In the study of exceptional Dehn fillings, two functions on slopes, called the Euclidean length on a horotorus and the Culler-Shalen norm, play important roles. In this paper, we investigate their relationship and establish two inequalities between them. As a byproduct, some bounds on the boundary slope diameter are given.

Euclidean lengths and the Culler-Shalen norms of slopes

TL;DR

This work investigates two slope-valued invariants in hyperbolic knot complements: the Euclidean length on a horotorus, length_T(r), and the Culler–Shalen norm, ||r||. It proves inequalities linking these quantities, notably a lower bound ||r|| ≥ (2/3) length_T(r) for certain knot exteriors and general two-slope configurations, and shows how length and norm bounds constrain boundary slopes and the boundary-slope diameter. The paper also derives general length–norm comparison results for numerical slopes, including equality cases under extremal slope conditions, and applies these to bound Diam(B_M). The figure-eight knot exterior serves as a concrete example demonstrating the sharpness and limitations of the bounds, with explicit formulas for length_T and ||r|| and a precise Diam(B_M) value.

Abstract

In the study of exceptional Dehn fillings, two functions on slopes, called the Euclidean length on a horotorus and the Culler-Shalen norm, play important roles. In this paper, we investigate their relationship and establish two inequalities between them. As a byproduct, some bounds on the boundary slope diameter are given.

Paper Structure

This paper contains 11 sections, 17 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Slopes
  4. Length on a horotorus
  5. Culler-Shalen norm
  6. Lower bound on Culler-Shalen norm
  7. Lengths, norms and numerical slopes
  8. Lengths of numerical slopes
  9. Norms of numerical slopes
  10. Boundary slope diameter
  11. Slopes for the figure-eight knot

Key Result

Theorem 1.1

Suppose that $M$ is the exterior of a hyperbolic two-bridge knot or the exterior of a $(-2,3,n)$-pretzel knot with $n$ odd and at least $7$ in $S^3$. Then holds for any slope $r$ on $\partial M$ and for any horotorus $T$. In particular, if $M$ is the exterior of a hyperbolic twist knot or the exterior of a $(-2,3,n)$-pretzel knot with $n$ odd and at least $7$ in $S^3$, then the same inequality ho

Figures (1)

  • Figure 1:

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem-LengthDistance']}
  • Lemma 2.2: Adams Ad
  • Lemma 2.3: Agol Ag
  • Proposition 2.4: Culler--Shalen CS04
  • Lemma 2.5
  • proof
  • ...and 20 more