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A Practical Algorithm for Knot Factorisation

Alexander He, Eric Sedgwick, Jonathan Spreer

Abstract

We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the $19$-crossing census. The main new ingredient in this work is an object that we call an "edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in $3$-manifold topology.

A Practical Algorithm for Knot Factorisation

Abstract

We present an algorithm for computing the prime factorisation of a knot, which is practical in the following sense: using Regina, we give an implementation that works well for inputs of reasonable size, including prime knots from the -crossing census. The main new ingredient in this work is an object that we call an "edge-ideal triangulation", which is what our algorithm uses to represent knots. As other applications, we give an alternative proof that prime knot recognition is in coNP, and present some new complexity results for triangulations. Beyond knots, our work showcases edge-ideal triangulations as a tool for potential applications in -manifold topology.

Paper Structure

This paper contains 28 sections, 21 theorems, 5 equations, 24 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Knots
  4. Triangulations of manifolds
  5. Normal surface theory
  6. Vertex normal surfaces and exchange annuli
  7. Crushing along normal surfaces
  8. Existence of suitable quad vertex normal spheres
  9. Tracking ideal edges under crushing
  10. Orbits of segments
  11. Trivial induced orbits and cycles of faces
  12. Induced orbit characterisation of quad vertex 2-spheres
  13. The effect of crushing on ideal loops
  14. Algorithm and implementation
  15. Correctness of the algorithm
  16. ...and 13 more sections

Key Result

theorem 4

There is an algorithm for Knot Factorisation that returns the prime factorisation of the input knot $K$ as a collection of edge-ideal triangulations.

Figures (24)

  • Figure 1: The unknot.
  • Figure 2: Two nontrivial knots (left and middle), and a connected sum of them (right).
  • Figure 4: The four triangle types.
  • Figure 5: The three quad types.
  • Figure 7: The link of a vertex is a normal surface built entirely from triangles.
  • ...and 19 more figures

Theorems & Definitions (37)

  • theorem 4
  • theorem 5
  • theorem 6: jaco1995algorithms
  • definition 1
  • definition 2: The crushing procedure burton2014crushing
  • lemma 1: Crushing lemma burton2014crushing
  • lemma 2
  • proof
  • lemma 3
  • proof
  • ...and 27 more