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A table of knotoids in $S^3$ up to seven crossings

Boštjan Gabrovšek, Paolo Cavicchioli

Abstract

We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids introduced by Turaev. We describe the methods used to enumerate diagrams, simplify them, and distinguish equivalence classes using a collection of invariants including the Kauffman bracket, the Arrow polynomial, the Affine index polynomial, the Mock Alexander polynomial, and the Yamada polynomial of the closure. We also investigate the chirality and rotational symmetries of these knotoids. Applications to protein entanglement illustrate the importance of such classifications.

A table of knotoids in $S^3$ up to seven crossings

Abstract

We present a complete classification of spherical knotoids with up to six crossings and conjecture that our classification up to seven crossings is complete. Our work extends the tradition of knot tabulation to the setting of knotoids introduced by Turaev. We describe the methods used to enumerate diagrams, simplify them, and distinguish equivalence classes using a collection of invariants including the Kauffman bracket, the Arrow polynomial, the Affine index polynomial, the Mock Alexander polynomial, and the Yamada polynomial of the closure. We also investigate the chirality and rotational symmetries of these knotoids. Applications to protein entanglement illustrate the importance of such classifications.
Paper Structure (23 sections, 2 equations, 11 figures, 2 tables)

This paper contains 23 sections, 2 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Knotoid invariants
  3. Bracket polynomial.
  4. Arrow polynomial.
  5. Affine index polynomial.
  6. Mock Alexander polynomial.
  7. Yamada polynomial of the closure.
  8. Prime knotoids and symmetries of knotoids
  9. Chirality.
  10. Rotatability.
  11. Knotoid notation
  12. PD notation.
  13. EM notation.
  14. Methodology
  15. Step 1: Generation of candidate diagrams.
  16. ...and 8 more sections

Figures (11)

  • Figure 1: Reidemeister moves of knotoid diagrams.
  • Figure 2: Forbidden moves
  • Figure 3: We obtain a knotoid representing a protein's backbone $P$ by placing the protein in a large enough sphere and projecting it to the tangent plane $\Sigma$ or to the sphere directly.
  • Figure 4: The closure of a knotoid.
  • Figure 5: Structural types of knotoids.
  • ...and 6 more figures