Duality in Non-polyhedral Bodies Part I: Polyliner

Abstract

Duality occurs in pairs of polyhedra, for example between the icosahedron and the dodecahedron, between the cube and the octahedron, and self-dually in the tetrahedron. In this paper, the principle of duality is generalized and linked to the construction of alternating knots. In a polyhedron, each surface has at least three vertices and there are at least four surfaces. There is always a dual polyhedron. From both, an alternating knot (or link) can be constructed, one that has as many crossing points as the polyhedron has edges. It turns out that this result also applies to bodies whose surfaces have less than three vertices and which consist of less than four faces. The resulting bodies can not be assembled like polyhedra from flat faces. In the following, they will be referred to as “polyliner”. If each facet has at least two vertices, an alternating knot can be constructed as before. If facets with only one point are present in the polyliner, the construction of the associated knot at this point results in a loop that can be unknotted. If the disentangling is not done, but the crossing point is maintained, then the resulting spatial curves can be cataloged according to their topology.