The Affine Index Polynomial and the Sawollek Polynomial

TL;DR

This paper explores the relationship between the Affine Index Polynomial and the Sawollek Polynomial in virtual knot theory, providing a new approach and a concise proof of Mellor's theorem connecting the two invariants.

Contribution

It introduces a new basis for analyzing the relationship between the Affine Index Polynomial and the Sawollek Polynomial, and offers a simplified proof of Mellor's extraction theorem.

Findings

Established a new framework for relating the two polynomials.

Provided a concise proof of Mellor's theorem.

Enhanced understanding of virtual knot invariants.

Abstract

The purpose of this paper is to give a new basis for examining the relationships of the Affine Index Polynomial and the Sawollek Polynomial. Blake Mellor has written a pioneering paper showing how the Affine Index Polynomial may be extracted from the Sawollek Polynomial. The Affine Index Polynomial is an elementary combinatorial invariant of virtual knots. The Sawollek polynomial is a relative of the classical Alexander polynomial and is defined in terms of a generalization of the Alexander module to virtual knots that derives from the so-called Alexander Biquandle. The present paper constructs the groundwork for a new approach to this relationship, and gives a concise proof of the basic Theorem of Mellor extracting the Affine Index Polynomial from the Sawollek Polynomial.