  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 paper 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 have
information "how many times a half of the crossing of the knot in $S_{g} \times
S^{1}$ rotates along $S^{1}$", we call it labels of crossings. In the end of
the present paper we extend this notion more generally and discuss its
geometrical meaning. This paper follows from \cite{Kim}.
