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Tangle decompositions of alternating link complements

Joel Hass, Abigail Thompson, Anastasiia Tsvietkova

Abstract

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produce a prime alternating link, if summed correctly with respect to the alternating property. Given a prime alternating link, we seek to understand whether it can be decomposed into two prime tangles each of which is alternating. We refine results of Menasco and Thistlethwaite to show that if such a decomposition exists either it is visible in an alternating link diagram or the link is of a particular form, which we call a pseudo-Montesinos link.

Tangle decompositions of alternating link complements

Abstract

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produce a prime alternating link, if summed correctly with respect to the alternating property. Given a prime alternating link, we seek to understand whether it can be decomposed into two prime tangles each of which is alternating. We refine results of Menasco and Thistlethwaite to show that if such a decomposition exists either it is visible in an alternating link diagram or the link is of a particular form, which we call a pseudo-Montesinos link.

Paper Structure

This paper contains 5 sections, 7 theorems, 7 figures.

Table of Contents

  1. Overview
  2. Conway spheres and standard position
  3. Decomposition into two prime alternating tangles
  4. Tangle decompositions of alternating braids
  5. Acknowledgements

Key Result

Lemma 2.1

Suppose $F$ is a closed essential 4-punctured sphere in the complement $S^3-L$ of a prime alternating non-split link $L$, i.e., $F$ is a Conway sphere for $L$. Then $F$ can be placed in standard position relative to $S^2_+ \cup S^2_-$ so that

Figures (7)

  • Figure 1:
  • Figure 2: Constructing a pseudo-Montesinos link.
  • Figure 3: Two $PSPS$ curves in a link diagram.
  • Figure 4: Diagrams of alternating braids
  • Figure 5: In grey, a curve that is monotone (left) and two curves that are not (center, right).
  • ...and 2 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • Remark 2.2
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Corollary 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 4 more