Knots in $S_{g} \times S^{1}$, their essential diagrams and virtual knots

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 $L = K\sqcup T$ 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.