Classification of knots in T x I with at most 4 crossings

TL;DR

This paper systematically classifies all knots in the thickened torus with diagrams having up to four crossings, using a comprehensive process and polynomial invariants to ensure completeness and distinction.

Contribution

It provides the first complete enumeration of such knots with up to four crossings, including new techniques for classification and proof of completeness.

Findings

Complete table of knots in T x I with ≤4 crossings

Introduction of new classification techniques

Use of generalized Kauffman polynomial for knot distinction

Abstract

We compose the table of knots in the thickened torus T x I having diagrams with at most 4 crossings. The knots are constructed by the three-step process. First we list regular graphs of degree 4 with at most 4 vertices, then for each graph we enumerate all corresponding knot projections, and after that we construct the corresponding minimal diagrams. Several known and new tricks made it possible to keep the process within reasonable limits and offer a rigorous theoretical proof of the completeness of the table. For proving that all knots are different we use a generalized version of the Kauffman polynomial.