Knots in S_g × S¹ and winding parities

2021/12/16 by Seongjeong Kim, Kim, Seongjeong
Computer Science · Engineering · Mathematics · #3D Shape Modeling and Analysis #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2112.08707

openalex publication_date 2021/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A virtual knot, which is one of generalizations of knots in ℝ³ (or S³), is, roughly speaking, an embedded circle in thickened surface S_g × I. In this paper we will discuss about knots in 3 dimensional S_g × S¹. We introduce basic notions for knots in S_g × S¹, for example, diagrams, moves for diagrams and so on. For knots in S_g × S¹ technically we lose over/under information, but we have information "how many times a half of the crossing of the knot in S_g × S¹ rotates along S¹", 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 \citeKim.

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