Alexander invariants for virtual knots

TL;DR

This paper introduces the virtual knot group and associated Alexander invariants, including polynomials and their normalized forms, which help analyze virtual knots and estimate their virtual crossing numbers.

Contribution

It constructs the virtual knot group and defines new invariants like the virtual Alexander polynomial and its twisted version, linking them to existing polynomials and providing skein relations.

Findings

The virtual Alexander polynomial is closely related to the generalized Alexander polynomial.

Normalized invariants satisfy skein formulas.

Bounds on virtual crossing numbers are derived from these invariants.

Abstract

Given a virtual knot $K$, we construct a group $VG_K$ called the virtual knot group, and we use the elementary ideals of $VG_K$ to define invariants of $K$ called the virtual Alexander invariants. For instance, associated to the $k=0$ ideal is a polynomial $H_K(s,t,q)$ in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial $G_K(s,t)$ introduced by Sawollek, Kauffman-Radford, and Silver-Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot $K$ and a representation $\varrho \colon VG_K \to GL_n(R)$, and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.