On links in $S_g\times S^1$ and its invariants

A virtual knot, which is one of generalizations of knots in ${\mathbb{R}}^3$ (or $S^3$), is, roughly speaking, an embedded circle in thickened surface $S_g\times I$. In this talk we will discuss about knots in 3 dimensional $S_g\times S^1$. We introduce basic notions for knots in $S_{g} \times S^{1}$,forexample,diagrams,movesfordiagramsandsoon\mathrm{.}Forknotsin$S_{g} \times S^{1}$ technically we lose over/under information, but we will have information how many times the knot rotates along $S^1$. We will discuss the geometric meaning of the rotating information and how to construct invariants by using the “rotating” information.