Knots in $S_g\times S^1$, their essential diagrams and virtual knots

Abstract

In \cite{Kim} it is shown that knots in $S_g\times S^1$ can be presented by virtual diagrams with a decoration, so called, {\em double lines}. In this paper, we study the essential diagram for each knot in $S_g\times S^1$, which has the minimal number of double lines. We prove that virtual knot theory is embedded in the theory of knots in $S_g\times S^1$. In the same time, one can obtain knots in $S^2\times S^1$ from 2-component links $L=K\bigsqcup T$ where $T$ is a trivial knot. By using knots in $S^2\times S^1$, we study the minimality and separability of such classical links.