A note on coverings of virtual knots

TL;DR

This paper demonstrates that for any finite set of virtual knots, there exists a single virtual knot whose various coverings match the set, showcasing the flexibility of virtual knot coverings.

Contribution

It proves the existence of a virtual knot with prescribed coverings for specific indices, expanding understanding of virtual knot structures.

Findings

Constructs a virtual knot with specified coverings

Shows the flexibility of the covering operation

Provides a method to realize any finite set of knots as coverings

Abstract

For a virtual knot $K$ and an integer $r\geq 0$, the $r$-covering $K^{(r)}$ is defined by using the indices of chords on a Gauss diagram of $K$. In this paper, we prove that for any finite set of virtual knots $J_0,J_2,J_3,\dots,J_m$, there is a virtual knot $K$ such that $K^{(r)}=J_r$ $(r=0\mbox{ and }2\leq r\leq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.