A note on coverings of virtual knots

For a virtual knot $K$ and an integer $rgeq 0$, 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 $K^{(r)}=J_r$ $(r=0mbox{ and }2leq rleq m)$, $K^{(1)}=K$, and otherwise $K^{(r)}=J_0$.