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Crossing Numbers of Knots on Closed Surfaces

Makoto Ozawa

Abstract

Let $c(K;F)$ denote the surface crossing number of a knot $K$ with respect to a closed surface $F \subset S^{3}$. We establish the lower bound \[ c(K;F) \ge 2\bigl(t(K)-δ(F)\bigr)+1, \] where $t(K)$ is the tunnel number of $K$ and $δ(F)$ is the Heegaard deficiency of $F$. In particular, for any fixed closed surface $F$, the surface crossing number $c(K;F)$ is unbounded over all knots $K$. Furthermore, we construct a family of knots $K_m$ demonstrating that $c(K_m;F) = Θ(t(K_m))$, which shows that this lower bound is asymptotically sharp.

Crossing Numbers of Knots on Closed Surfaces

Abstract

Let denote the surface crossing number of a knot with respect to a closed surface . We establish the lower bound where is the tunnel number of and is the Heegaard deficiency of . In particular, for any fixed closed surface , the surface crossing number is unbounded over all knots . Furthermore, we construct a family of knots demonstrating that , which shows that this lower bound is asymptotically sharp.
Paper Structure (35 sections, 13 theorems, 28 equations, 1 figure)

This paper contains 35 sections, 13 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Main Results
  4. Organization of the Paper
  5. Preliminaries
  6. Handlebodies, compression bodies, and Heegaard genus
  7. Knots and diagrams on closed surfaces
  8. The surface bridge number
  9. The surface ascending number
  10. Monotonic behavior
  11. Tunnel number
  12. Heegaard splittings determined by a closed surface
  13. The $h$-genus
  14. Main Results
  15. A general lower bound
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $F\subset S^3$ be a closed surface. Then every knot $K\subset S^3$ satisfies: where $t(K)$ is the tunnel number of $K$, and $\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.

Figures (1)

  • Figure 1: A schematic cross-section of the amalgamation of Heegaard splittings. The amalgamated surface $F'$ (dashed blue) is obtained from $F$ by tubing along the 1-handles of the compression bodies $C_1$ and $C_2$. The knot $K$ (red) is in a bridge position with respect to $F$ and is isotoped to be disjoint from the 1-handles, preserving its bridge arcs in the resulting handlebodies bounded by $F'$.

Theorems & Definitions (36)

  • Theorem 1.1: Fundamental inequality
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Surface crossing number
  • Remark 2.4
  • Definition 2.5: Diagrammatic definition
  • Definition 2.6: Geometric definition via height function
  • Definition 2.7
  • Proposition 2.8
  • proof
  • ...and 26 more