Isotopy classes of 3-periodic net embeddings

TL;DR

This paper classifies isotopy classes of 3-periodic nets, introduces a new methodology using linear graph knots on the 3-torus, and provides detailed classifications for specific net families and symmetries.

Contribution

It offers the first comprehensive classification of 3-periodic net isotopy classes and introduces linear graph knots as a novel analytical tool.

Findings

Enumerated 25 isotopy classes of depth 1 nets with a single vertex quotient.

Classified embeddings of n-fold copies of pcu with parallel components.

Determined maximal symmetry periodic isotopes of certain net embeddings.

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.