A characterization of virtual knots as knots in $S_{g} \times S^{1}$

In this talk we will show that virtual knots are embedded in the set of knots in $S_{g} \times S^{1}$. We will also provide a sufficient condition for knots in $S_{g} \times S^{1}$ to have virtual knot diagrams. Based on this, we derive a sufficient condition for 2-component classical links to be separable.