Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

07 July 2020, Version 1
This content is a preprint and has not undergone peer review at the time of posting.

Abstract

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.

Keywords

Isogonal weaves
Polycatenanes
Knots
Weaves
Stick transitivity

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