Piecewise-linear embeddings of knots and links with rotoinversion symmetry

Tangled piecewise-linear embeddings of trivalent graphs

A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent graphs with 6 vertices (two found) and 8 vertices (45 found). Symmetrical tangles of the 10-vertex Petersen graph and the 20-vertex Desargues graph are also described. Extensions to periodic tangles are indicated. These are all interesting and viable targets for molecular synthesis.

On Borromean links and related structures

The creation of knotted, woven and linked molecular structures is an exciting and growing field in synthetic chemistry. Presented here is a description of an extended family of structures related to the classical `Borromean rings', in which no two rings are directly linked. These structures may serve as templates for the designed synthesis of Borromean polycatenanes. Links of n components in which no two are directly linked are termed `n-Borromean' [Liang & Mislow (1994). J. Math. Chem. 16, 27-35]. In the classic Borromean rings the components are three rings (closed loops). More generally, they may be a finite number of periodic objects such as graphs (nets), or sets of strings related by translations as in periodic chain mail. It has been shown [Chamberland & Herman (2015). Math. Intelligencer, 37, 20-25] that the linking patterns can be described by complete directed graphs (known as tournaments) and those up to 13 vertices that are vertex-transitive are enumerated. In turn, these lead to ring-transitive (isonemal) n-Borromean rings. Optimal piecewise-linear embeddings of such structures are given in their highest-symmetry point groups. In particular, isonemal embeddings with rotoinversion symmetry are described for three, five, six, seven, nine, ten, 11, 13 and 14 rings. Piecewise-linear embeddings are also given of isonemal 1- and 2-periodic polycatenanes (chains and chain mail) in their highest-symmetry setting. Also the linking of n-Borromean sets of interleaved honeycomb nets is described.