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A robot that unknots knots

Connie On Yu Hui, Dionne Ibarra, Louis H. Kauffman, Emma N. McQuire, Gabriel Montoya-Vega, Sujoy Mukherjee, Corbin Reid

TL;DR

The paper introduces a memoryless robot that, when run on knot diagrams, yields ascending diagrams and supports a new combinatorial proof that ascending/descending diagrams are unknots. It derives a quadratic upper bound $$(7C+1)C$$ on the number of Reidemeister moves needed to simplify such diagrams and provides a loop detour framework to realize monotone simplifications. The work also extends to links, offers an alternative proof for the simplification theorem, and draws interdisciplinary connections to electrical networks and DNA topology. The results advance understanding of diagrammatic unknotting and offer constructive methods with potential applications in graph-based representations and biological topology. The framing and detour machinery give a new toolkit for analyzing and simplifying knot diagrams with clear complexity bounds.

Abstract

Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending diagram, and we provide a new combinatorial proof that every ascending or descending knot diagram can be transformed into the zero-crossing unknot diagram. Using the machinery developed from the combinatorial proof, we show that the minimal number of Reidemeister moves required for such a transformation is bounded above by (7C+1)C if the diagram has C crossings. Moreover, we provide a new alternative proof that there exist sequences of Reidemeister moves that do not increase the number of crossings and transform ascending or descending knot diagrams into zero-crossing unknot diagrams.

A robot that unknots knots

TL;DR

The paper introduces a memoryless robot that, when run on knot diagrams, yields ascending diagrams and supports a new combinatorial proof that ascending/descending diagrams are unknots. It derives a quadratic upper bound on the number of Reidemeister moves needed to simplify such diagrams and provides a loop detour framework to realize monotone simplifications. The work also extends to links, offers an alternative proof for the simplification theorem, and draws interdisciplinary connections to electrical networks and DNA topology. The results advance understanding of diagrammatic unknotting and offer constructive methods with potential applications in graph-based representations and biological topology. The framing and detour machinery give a new toolkit for analyzing and simplifying knot diagrams with clear complexity bounds.

Abstract

Consider a robot that remembers only the starting position and walks along a knot once on a knot diagram, switching every undercrossing it meets until it returns to the starting position. We observe that the robot produces an ascending diagram, and we provide a new combinatorial proof that every ascending or descending knot diagram can be transformed into the zero-crossing unknot diagram. Using the machinery developed from the combinatorial proof, we show that the minimal number of Reidemeister moves required for such a transformation is bounded above by (7C+1)C if the diagram has C crossings. Moreover, we provide a new alternative proof that there exist sequences of Reidemeister moves that do not increase the number of crossings and transform ascending or descending knot diagrams into zero-crossing unknot diagrams.

Paper Structure

This paper contains 13 sections, 31 theorems, 19 equations, 27 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. An upper bound on the minimal number of Reidemeister moves for ascending diagrams
  3. Simplification of ascending diagrams
  4. Interdisciplinary connections
  5. Acknowledgements
  6. Preliminaries
  7. The unknotting robot
  8. The unknotting robot unlinks links
  9. An upper bound on the number of Reidemeister moves
  10. A new alternative proof for the simplification theorem
  11. Interdisciplinary connections
  12. Electrical networks
  13. DNA topology

Key Result

Theorem 1.2

The robot in Algorithm Alg:Robot produces an ascending diagram.

Figures (27)

  • Figure 1: Illustration of a projection of an ascending diagram viewed from above, with starting point marked. If one pulls up the starting point, the diagram unknots.
  • Figure 2: Action of robot at overcrossing (top) and undercrossing (bottom). Arrow shows direction of robot's travel.
  • Figure 3: Illustration of a detour move.
  • Figure 4: A (6,4)-tangle detour move. The detour arc is also shown.
  • Figure 5: An imaginary example that helps explain the proof of Lemma \ref{['Lem:NoCommonEdge']}: A $4$-valent subgraph with a $13$-sided polygon $P$. Side 5 and Side 9 share the same edge $E$, which connects a $3$-sided polygon $P_3$ and an $8$-sided polygon $P_8$. A planar $4$-valent graph cannot have such a subgraph because the number of internally incident edges for $P_3$ can only be even, not odd.
  • ...and 22 more figures

Theorems & Definitions (74)

  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 64 more