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A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots

Xiao Chen, Wujie Shen

Abstract

This paper focuses on the Kakimizu complex of a hyperbolic knot $K$. We define a complex $IS_\ell(K)$ to study incompressible Seifert surfaces of genus at most $\ell$, and prove that it is connected and that its diameter admits a linear upper bound in terms of $\ell$. As a corollary, we show that the diameter of the Kakimizu complex $MS(K)$ of a hyperbolic knot grows linearly with the genus $g$, confirming a conjecture of Sakuma--Shackleton. More precisely, it is bounded above by $6g-4$.

A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots

Abstract

This paper focuses on the Kakimizu complex of a hyperbolic knot . We define a complex to study incompressible Seifert surfaces of genus at most , and prove that it is connected and that its diameter admits a linear upper bound in terms of . As a corollary, we show that the diameter of the Kakimizu complex of a hyperbolic knot grows linearly with the genus , confirming a conjecture of Sakuma--Shackleton. More precisely, it is bounded above by .

Paper Structure

This paper contains 13 sections, 22 theorems, 41 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Kakimizu's complexes
  4. Atoroidal knots and hyperbolic knots
  5. Intersection of Two Seifert Surfaces
  6. A Technical Lemma
  7. Infinite Cyclic Cover
  8. Definition of infinite cyclic cover
  9. Infinite cyclic cover decomposition
  10. Kakimizu's characterisation
  11. Annulus Surgery
  12. Central Zone
  13. Proof of \ref{['main thm: Hyperbolic Knot Linear Bound']}

Key Result

Theorem 1.1

Suppose $K$ is an atoroidal knot in $\mathbb{S}^3$ of genus $g$. Then

Figures (8)

  • Figure 1: The pair $(X_1, A_1)$ can be given a product structure $\left(A_1 \times[0,1], A_1 \times\{0\}\right)$.
  • Figure 2: A lift of the Seifert surface $S_2$.
  • Figure 3: The relative position of the lifts of $S_1$ and $S_2$.
  • Figure 4: $I$-shaped and $U$-shaped annuli.
  • Figure 5: Annulus surgery.
  • ...and 3 more figures

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Kakimizu complex, MR1177053
  • Definition 2.2: Incompressible Seifert surface complex, MR1177053
  • Definition 2.3: $\ell$-Kakimizu complex
  • Remark 2.4
  • Definition 2.5: Natural distance, MR2531146
  • Definition 2.6: Diameter
  • Definition 2.7: Incompressible torus
  • Definition 2.8: Essential torus
  • ...and 49 more