Links in $S_{g} \times S^{1}$ and invariants derived from their liftings

Virtual links can be considered as links in a thickened surface $S_{g} \times I$. The three dimensional topology of the ambient space $S_{g} \times I$ plays important role for virtual links theory. Now, we studied links in $S_{g} \times S^{1}$. The ambient space has a topology from the $S^{1}$ instead of $I$. Moreover, it can be connected to usual virtual links by the lifting of the knots in $S_{g} \times S^{1}$ along the projection map $p$ from $S_{g} \times \mathbb{R}$ to $S_{g} \times S^{1}$. By lifting them along $p$, we get a link in $S_{g} \times S^{1}$ with infinite components or a string link with $n$-components. In this talk, we will introduce the diagram for links in $S_{g} \times S^{1}$ and the moves for the diagrams. We show that it is sufficient to study the diagrams for studying links in $S_{g} \times S^{1}$. And we will introduce an invariant for string links and links. Finally, we apply the invariant to knots in $S_{g} \times S^{1}$ to define an invariant for them.