Knots in $S_{g} \times S^{1}$ and winding parities

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}.

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