Virtual shadow modules and their link invariants

TL;DR

This paper introduces algebraic modules linked to virtual biracks and shadows to enhance knot invariants, capable of detecting orientation reversal and distinguishing knots beyond traditional polynomials.

Contribution

It develops a new algebraic framework for virtual shadow modules and extends existing invariants to the virtual and twisted virtual cases.

Findings

Invariants can detect orientation reversal.

Invariants are independent of the knot group and certain polynomials.

Twisted invariants distinguish knots beyond twisted Jones polynomial.

Abstract

We introduce an algebra Z[X,S] associated to a pair (X,S) of a virtual birack X and X-shadow S. We use modules over Z[X,S] to define enhancements of the virtual birack shadow counting invariant, extending the birack shadow module invariants to virtual case. We repeat this construction for the twisted virtual case. As applications, we show that the new invariants can detect orientation reversal and are not determined by the knot group, the Arrow polynomial and the Miyazawa polynomial, and that the twisted version is not determined by the twisted Jones polynomial.