Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

3-decompositions of genus two handlebody-knots

Makoto Ozawa, Yi-Sheng Wang

Abstract

We investigate the class of $3$-decomposable genus two handlebody-knots and provide a complete classification of essential annuli in their exteriors. We introduce the notion of $τ$- and $ρ$-tangles and good rectangles and annuli. By classifying $τ$- and $ρ$-tangles whose exteriors admit a good rectangle or annulus, we categorize atoroidal $3$-decomposable genus two handlebody-knots into distinct classes, based on the number of essential annuli. As an application, the hyperbolicity of all genus two handlebody-knots with up to six crossings are determined, and numerous hyperbolic handlehody-knots with seven crossings identified. Furthermore, obstructions for a handlebody-knot to be $3$-decomposable are constructed with explicit examples provided.

3-decompositions of genus two handlebody-knots

Abstract

We investigate the class of -decomposable genus two handlebody-knots and provide a complete classification of essential annuli in their exteriors. We introduce the notion of - and -tangles and good rectangles and annuli. By classifying - and -tangles whose exteriors admit a good rectangle or annulus, we categorize atoroidal -decomposable genus two handlebody-knots into distinct classes, based on the number of essential annuli. As an application, the hyperbolicity of all genus two handlebody-knots with up to six crossings are determined, and numerous hyperbolic handlehody-knots with seven crossings identified. Furthermore, obstructions for a handlebody-knot to be -decomposable are constructed with explicit examples provided.
Paper Structure (23 sections, 22 theorems, 5 equations, 16 figures)

This paper contains 23 sections, 22 theorems, 5 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decomposition of spatial graphs and handlebody-knots
  4. Essential disks, annuli and tori
  5. Genus two case
  6. Weak decomposition
  7. Rectangles in $3$-tangle exteriors
  8. Decomposition
  9. Standard disks in $B_L$
  10. Rectangles in $(B_R,G_R)$
  11. Rational $3$-tangles
  12. Torus $\rho$-tangle
  13. Rectangles in $\tau$-tangle exteriors
  14. Rectangles of type I
  15. Rectangles of type II
  16. ...and 8 more sections

Key Result

Lemma 2.1

An $n$-tangle decomposition $(B_1,G_1)\cup_S (B_2,G_2)$ of a spacial graph $\Gamma$ is essential if and only if both $(B_i,G_i)$, $i=1,2$, are essential.

Figures (16)

  • Figure 1: Intersection $S\cap V$.
  • Figure 2: $\tau$- and $\rho$-tangles.
  • Figure 3:
  • Figure 4: Standard disks.
  • Figure 5: Rational $3$-tangles.
  • ...and 11 more figures

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Definition 2.3
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 38 more