Triple-crossing projections, moves on knots and links and their minimal diagrams

TL;DR

This paper introduces a systematic method for generating and classifying prime knots and links with minimal triple-crossing projections, extending existing tables and reducing diagrammatic moves through a new move type.

Contribution

It presents a novel systematic approach for generating minimal triple-crossing projections and classifies knots and links with triple-crossing number up to five, introducing a new diagrammatic move.

Findings

Classified all prime knots and links with triple-crossing number ≤ 4.

Extended the table of knots with triple-crossing number 5.

Derived a minimal set of moves connecting diagrams of the same knot.

Abstract

In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.