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Generating Strictly Controlled Stimuli for Figure Recognition Experiments

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Isotopy classes for 3-periodic net embeddings.

Stephen C Power¹, Igor A Baburin², Davide M Proserpio³

¹Department of Mathematics and Statistics, Lancaster University, Bailrigg, Lancaster LA1 1SQ, United Kingdom.

Acta Crystallographica. Section A, Foundations and Advances

|May 2, 2020

Summary

This summary is machine-generated.

This study defines entangled embedded periodic nets and crystal frameworks, classifying their types and enumerating specific classes. It introduces linear graph knots for analyzing spatial embeddings and their symmetries.

Keywords:

coordination polymers crystallographic frameworks embedded nets isotopy types periodic nets

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Area of Science:

Crystallography
Topology
Materials Science

Background:

Periodic nets and crystal frameworks are fundamental structures in materials science and crystallography.
Understanding their topological and geometric properties is crucial for designing new materials.
Existing classifications often lack comprehensive analysis of isotopy types and spatial embeddings.

Purpose of the Study:

To define and classify entangled embedded periodic nets and crystal frameworks based on dimension, homogeneity, adjacency depth, and periodic isotopy type.
To enumerate periodic isotopy classes for specific families of embedded nets, particularly those with small quotient graphs.
To introduce a novel methodology for analyzing spatial embeddings using linear graph knots on a 3-torus.

Main Methods:

Formal definition of entangled embedded periodic nets and crystal frameworks.
Classification based on dimension type, homogeneity type, adjacency depth, and periodic isotopy type.
Enumeration of periodic isotopy classes for depth-1 embedded nets with single-vertex quotient graphs.
Classification of embeddings of n-fold copies of pcu (primitive cubic unit cell).
Introduction of linear graph knots on the flat 3-torus for spatial embedding analysis.

Main Results:

Periodic isotopy classifications for various embedded net families with small quotient graphs.
Enumeration of 25 periodic isotopy classes for depth-1 embedded nets with a single-vertex quotient graph.
Classification of specific embeddings of n-fold pcu copies and their maximal symmetry periodic isotopes.
Development of the linear graph knot methodology for analyzing spatial embeddings of quotient graphs.

Conclusions:

The study provides a systematic classification of entangled embedded periodic nets and crystal frameworks.
The introduced linear graph knot methodology offers a new tool for understanding the topology of crystal structures.
These findings contribute to the fundamental understanding of periodic structures and their spatial arrangements.