The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space

We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide.

Isogonal piecewise-linear embeddings of 1-periodic knots and links, and related 2-periodic chain-link and knitting patterns

Families of 1- and 2-periodic knots and weavings that have isogonal (vertex-transitive) piecewise-linear embeddings are described. In these structures there is just one thread, or multiple threads with parallel or collinear axes. The principal structures are a large family of 1-periodic knots and related multi-thread infinite links, knitting patterns and chain-link weaving. The relevance to synthetic chemistry is described in terms of targets for designed synthesis such as mechanically interlocked polymers.