On links in $S_{g} \times S^{1}$ and its invariants

Abstract

A virtual knot, which is one of generalizations of knots in R 3 (or S 3 ), is, roughly speaking, an embedded circle in thickened surface Sg × I. In this talk we will discuss about knots in 3 dimensional Sg × S 1 . We introduce basic notions for knots in Sg × S 1 , for example, diagrams, moves for diagrams and so on. For knots in Sg × S 1 technically we lose over/under information, but we will have information how many times the knot rotates along S 1 . We will discuss the geometric meaning of the rotating information and how to construct invariants by using the "rotating" information.

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

“…In [1], the author constructed knots in S g ×S 1 and local moves. In [2], the author defined "labels" of crossings of knots in S g × S 1 and its applications.…”