On the determinant of checkerboard colorable virtual knots

TL;DR

This paper extends classical knot invariants to checkerboard colorable virtual knots, showing that their determinant modulo 8 is classified by a specific coefficient in an extended polynomial.

Contribution

It introduces a new classification of the determinant modulo 8 for checkerboard colorable virtual knots using an extended polynomial invariant.

Findings

Determinant modulo 8 is classified by the z^2 coefficient in the ascending polynomial.

Extension of the Conway polynomial to virtual knots.

Provides a classification similar to classical knots using new polynomial invariants.

Abstract

For classical knots, Murasugi showed that the determinant modulo $8$ is classified by the Arf invariant. Boden and Karimi introduced a determinant for checkerboard colorable virtual knots. We prove that this determinant modulo $8$ is classified by the coefficient of $z^2$ in the ascending polynomial, an extension of the Conway polynomial for classical knots.