Isotopy classes for 3-periodic net embeddings

Power, Stephen and Baburin, Igor and Proserpio, Davide (2020) Isotopy classes for 3-periodic net embeddings. Acta Crystallographica Section A: Foundations and Advances, 76 (3). pp. 275-301. ISSN 2053-2733

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Abstract

Entangled embedded periodic nets and crystal frameworks are defined, along with their {dimension type}, {homogeneity type}, {adjacency depth} and {periodic isotopy type}. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of depth 1 embedded nets with a single vertex quotient graph. Additionally, we classify embeddings of n-fold copies of {pcu} with all connected components in a parallel orientation and n vertices in a repeat unit, and determine their maximal symmetry periodic isotopes. We also introduce the methodology of linear graph knots on the flat 3-torus [0, 1)^3. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.