Knots in Sg×S1 and winding parities.

In knot theory not only classical knots, which are embedded circles in S 3 up to isotopy, but also knots in other 3-manifolds are interesting for mathematicians. In particular, virtual knots, which are knots in thickened surface S g × [ 0 , 1 ] with an orientable surface S g of genus g , are studied and they provide interesting properties. One of the famous tools to study virtual knots is parity defined by Manturov, by using which many invariants for classical knots can be nontrivially extended to invariants for virtual knots. In this paper, we are interested in knots in S g × S 1 . Isotopy classes of knots in S g × S 1 can be presented by using diagrams on plane and local moves, but one can expect that we lose over/under information. But we have information "how many times a half of the crossing of the knot in S g × S 1 rotates along S 1 ", and we define it labels of crossings. We extend labels to the notion of winding parity and properties of it are studied. In the end of the paper, we study classifications of knots in S g × S 1 with small number of crossings by using the winding parity. [ABSTRACT FROM AUTHOR]