Defining Curve Behaviors on the Surface of a Möbius Strip with One Boundary using the Double Cover

The Möbius strip in topology is a non-orientable 1-sided surface that can only be expressed in 3D or a higher dimension. Although the Möbius strip is famous as being a 3D surface with one side, and in the topological world as being the basis of the Klein bottle, a Möbius strip with a boundary is much less studied. In this paper, I explore the surface that I have defined. To help improve the understanding of this surface, I needed to create a workable map design to represent the Möbius strip surface in 2D.  After experimenting with multiple prototypes of 2D maps, I came up with an efficient mapping model that allows simple and comprehensible representations of curve behaviors on the surface of a Möbius strip with one boundary while conserving the properties of the surface itself.  The purpose of this paper is to explore the simple curve types and directions to provide proof of the 3 directions possible on the surface as well as the 4 basic curve types using the rules of string topology.

​

FIELD: String Topology​