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

A virtual knot, which is one of generalizations of knots in $R^3$ (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 $S_g \times S^1$, for example, diagrams, moves for diagrams and so on. For knots in $S_g \times S^1$ 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.