Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes
We describe mathematical knots and links as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ("vertex" transitive). Corner- and stick-transitive structures are termed regular. We find no regular knots. Regular links are cubic or icosahedral and a complete account of these is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. We note the relevance of this work to materials- and bio-chemistry.