Knots in $S_g\times S^1$ and winding parities

Abstract

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}.

This paper follows the paper arXiv:2104.08573 [math.GT]