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The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings

Rhea Palak Bakshi, Benjamin A. Burton, Huizheng Guo, Dionne Ibarra, Gabriel Montoya-Vega, Sujoy Mukherjee, Józef H. Przytycki

Abstract

Yasutaka Nakanishi formulated the following conjecture in 1981: every link is 3-move equivalent to a trivial link. While the conjecture was proved for several specific cases, it remained an open question for over twenty years. In 2002, Mieczysław D{\c a}bkowski and the last author showed that it does not hold, in general. In this article, we prove the Montesinos-Nakanishi $3$-move conjecture for links with up to 19 crossings and, with the exception of six pairwise non-isotopic links including the Chen link and its mirror image, for links with 20 crossings. Our work completely classifies links up 20 crossings modulo $3$-moves. This work includes computational methods, including new code in Regina that generalises pre-existing knot functions to work with links.

The Montesinos-Nakanishi 3-move conjecture for links up to 20 crossings

Abstract

Yasutaka Nakanishi formulated the following conjecture in 1981: every link is 3-move equivalent to a trivial link. While the conjecture was proved for several specific cases, it remained an open question for over twenty years. In 2002, Mieczysław D{\c a}bkowski and the last author showed that it does not hold, in general. In this article, we prove the Montesinos-Nakanishi -move conjecture for links with up to 19 crossings and, with the exception of six pairwise non-isotopic links including the Chen link and its mirror image, for links with 20 crossings. Our work completely classifies links up 20 crossings modulo -moves. This work includes computational methods, including new code in Regina that generalises pre-existing knot functions to work with links.

Paper Structure

This paper contains 11 sections, 7 theorems, 31 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Method
  3. Conway's basic polyhedra
  4. Reductions and rearrangements of Conway's polyhedra
  5. Computational process
  6. Burnside group of a link
  7. Classical Burnside groups
  8. The core group of a link
  9. The Burnside group of a link from the core group
  10. 5-braids, the 3-move operation, and the third Burnside group
  11. Future directions

Key Result

Theorem 1.1

Figures (10)

  • Figure 1.1: The 3-move operation.
  • Figure 1.2: Chen's 20-crossing counterexample to the Montesinos-Nakanishi $3$-move conjecture, which is the closure of the five braid $(\sigma_2\sigma_1^{-1}\sigma_2\sigma_3\sigma_4^{-1})^4$.
  • Figure 2.1: 2-triangles configuration and $\frac{\pi}{3}$ positive (counter-clockwise) rotation.
  • Figure 2.2: Rotation by $\frac{\pi}{3}$ (clockwise direction).
  • Figure 2.3: Configurations that are 3-move reducible.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Theorem 1.1: Main Theorem
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 5 more