Liftings of knots in $S_{g} \times S^{1}$ and covering of virtual knots

TL;DR

This paper introduces a new method for constructing families of $m$-fold coverings of virtual knots that are mod $m$ almost classical, using knots in the product space $S_{g} imes S^{1}$, expanding the tools for virtual knot analysis.

Contribution

It presents a novel approach to generate $m$-fold coverings of virtual knots via knots in $S_{g} imes S^{1}$, complementing existing methods based on oriented cut points.

Findings

Provides a new construction method for mod $m$ almost classical virtual knots.

Connects virtual knot coverings with knots in surface product spaces.

Enhances understanding of virtual knot invariants through geometric constructions.

Abstract

A virtual link diagram is called {\em (mod $m$) almost classical} if it admits a (mod $m$) Alexander numbering. In \cite{BodenGaudreauHarperNicasWhite}, it is shown that Alexander polynomial for almost classical links can be defined by using the homology of the associated infinite cyclic cover. On the other hand, in \cite{NaokoKamada} an infinite family of $m$ fold covering over a virtual knot is constructed so that it is mod $m$ almost classical link for all $m$ by using oriented cut point. In this paper, another way to obtain a family of $m$-fold coverings over a given virtual knots, which are mod $m$ almost classical, by using knots in $S_{g} \times S^{1}$.