Simplified numerical form of universal finite type invariant of Gauss words

TL;DR

This paper simplifies the universal finite type invariant of Gauss words using matrix transformations, enabling explicit computation up to degree 6 and aiding in classifying Gauss words of ranks 4 and 5.

Contribution

It introduces a simplified, computationally implementable form of the universal finite type invariant for Gauss words, extending explicit calculations to higher degrees.

Findings

Universal finite type invariant explicitly computed for degrees 4, 5, and 6.

Complete classification of Gauss words of rank 4.

Partial classification of Gauss words of rank 5.

Abstract

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.