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

TL;DR

This paper provides a comprehensive classification of spherical knotoids with up to six crossings and extends the enumeration to seven crossings, using various invariants and symmetry analysis, with applications in protein entanglement.

Contribution

It offers the first complete classification of knotoids up to six crossings and a conjecture for seven crossings, extending knot tabulation methods to knotoids.

Findings

Complete classification of knotoids up to six crossings.

Conjecture on the completeness of seven crossings classification.

Analysis of symmetry properties and applications in biology.

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.