Analytic representation of generalized möbius-listing’s bodies and classification of links appearing after their Cut

Abstract

For more than almost 200 years the Möbius strip and its “mysterious” property attracts the attention of mathematicians. After a “complete cut” of this surface, one object appears, but already with a fourfold twist. The generalization of this phenomenon to figures of a more complex configuration led to an “unexpected” result: after the cut of the generalized Möbius-Listing body, more than two geometric shapes may appear. In this paper, we consider all possible cases of a complete cut of the generalized Möbius-Listing body with a regular hexagon as radial section. In early works, together with different colleagues, on the basis of importance, they separately examined the case of Möbius-Listing’s bodies with a radial section of regular 3, 4 and 5 angular figures. Also, cases of similar bodies with a radial section of convex regular two and three angular figures were considered separately. One possible application of these results is assumed in the description of the properties of the middle surfaces in the theory of elastic shells [14] (Vekua, Shell Theory: General Methods of Construction. Pitman Advanced Publishing Program, Boston, p. 287, 1985).