Tabulation of knots up to five triple-crossings and moves between oriented diagrams

TL;DR

This paper catalogs minimal diagrams of prime knots with up to five triple-crossings, introduces moves connecting such diagrams, and conjectures a link between triple-crossing number and Alexander polynomial breadth.

Contribution

It provides the first comprehensive tables of minimal triple-crossing diagrams for prime knots and establishes a set of moves for transforming these diagrams.

Findings

Tables of minimal diagrams for prime knots up to triple-crossing number five

A minimal generating set of moves connecting triple-crossing diagrams

A conjecture relating triple-crossing number to Alexander polynomial breadth

Abstract

We enumerate and show tables of minimal diagrams for all prime knots up to the triple-crossing number equal to five. We derive a minimal generating set of oriented moves connecting triple-crossing diagrams of the same oriented knot. We also present a conjecture about a strict lower bound of the triple-crossing number of a knot related to the breadth of its Alexander polynomial.