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Highly twisted diagrams

Nir Lazarovich, Yoav Moriah, Tali Pinsky

TL;DR

This work addresses when links with $3$-highly twisted, twist-reduced diagrams and at least two twist regions are hyperbolic. The authors replace geometric arguments with a combinatorial Euler characteristic analysis of taut surfaces in normal position, proving the complement is atoroidal and unannular and thus hyperbolic. A key contribution is the Euler-characteristic redistribution technique, which bounds the possible surface intersections and eliminates essential spheres and annuli. The results sharpen existing $6$-highly twisted criteria, demonstrate sharpness via $2$-highly twisted counterexamples, and yield corollaries on essential holed spheres in highly twisted link complements, broadening practical criteria for hyperbolicity in knot theory.

Abstract

We prove that the knots and links that admit a 3-highly twisted irreducible diagram with more than two twist regions are hyperbolic. This should be compared with a result of Futer-Purcell for 6-highly twisted diagrams. While their proof uses geometric methods our proof is achieved by showing that the complements of such knots or links are unannular and atoroidal. This is done by using a new approach involving an Euler characteristic argument.

Highly twisted diagrams

TL;DR

This work addresses when links with -highly twisted, twist-reduced diagrams and at least two twist regions are hyperbolic. The authors replace geometric arguments with a combinatorial Euler characteristic analysis of taut surfaces in normal position, proving the complement is atoroidal and unannular and thus hyperbolic. A key contribution is the Euler-characteristic redistribution technique, which bounds the possible surface intersections and eliminates essential spheres and annuli. The results sharpen existing -highly twisted criteria, demonstrate sharpness via -highly twisted counterexamples, and yield corollaries on essential holed spheres in highly twisted link complements, broadening practical criteria for hyperbolicity in knot theory.

Abstract

We prove that the knots and links that admit a 3-highly twisted irreducible diagram with more than two twist regions are hyperbolic. This should be compared with a result of Futer-Purcell for 6-highly twisted diagrams. While their proof uses geometric methods our proof is achieved by showing that the complements of such knots or links are unannular and atoroidal. This is done by using a new approach involving an Euler characteristic argument.
Paper Structure (12 sections, 23 theorems, 18 equations, 40 figures, 1 table)

This paper contains 12 sections, 23 theorems, 18 equations, 40 figures, 1 table.

Table of Contents

  1. introduction
  2. preliminaries
  3. Bubbles and twist regions.
  4. Surfaces in link complements.
  5. Normal position.
  6. Curves of intersection.
  7. Taut surfaces
  8. Euler Characteristic and curves of intersection
  9. Distributing Euler characteristic among curves
  10. Redistribution of Euler characteristic
  11. Atoroidal and unannular
  12. Essential holed spheres in highly twisted link complements

Key Result

Theorem A

Let $D(\mathcal{L})$ be a connected, prime, twist-reduced, $3$-highly twisted diagram of a link $\mathcal{L}$ with at least two twist regions, then $\mathcal{L}$ is hyperbolic.

Figures (40)

  • Figure 1: A non-hyperbolic link with a 2-highly twisted diagram.
  • Figure 2: A 3-highly twisted link diagram. The twist regions are the dashed rectangles.
  • Figure 3: The possible three types of intersection of $S$ with a twist box.
  • Figure 5: The six possibilities for a curve in $c\in\mathcal{C}_{2,0}\cup \mathcal{C}_{1,2}$
  • Figure 6: The five configurations of good curves which are not in $\mathcal{C}_{2,0}$
  • ...and 35 more figures

Theorems & Definitions (74)

  • Theorem A
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.4
  • Remark 3.5
  • Definition 3.6
  • ...and 64 more