Nets with collisions (unstable nets) and crystal chemistry

Area of Science:

• Crystallography
• Materials Science
• Network Theory

Background:

• Unstable nets, characterized by vertices with identical barycentric coordinates (collisions), are crucial in understanding complex network structures.
• Some unstable nets possess non-crystallographic symmetries, deviating from standard crystallographic principles.
• The study investigates embeddings of nets from real crystal structures, focusing on symmetry and node relationships.

Purpose of the Study:

• To analyze the properties of unstable nets, particularly those with crystallographic symmetries.
• To explore the relationship between node collisions, topological neighbors, and symmetry in net embeddings.
• To introduce and exemplify the concept of 'unforced' collisions in crystallographic nets.

Main Methods:

• Examination of net structures and their barycentric coordinate representations.
• Analysis of net embeddings with crystallographic symmetry.
• Classification of node relationships (linked or unlinked) in the context of collisions.
• Identification and characterization of non-crystallographic symmetries in unstable nets.

Main Results:

• Examples of nets from real crystal structures with crystallographic symmetry embeddings are presented.
• It is shown that colliding nodes can be either topological neighbors or not, and links can coincide.
• A specific crystallographic net of girth 16 with barycentric coordinate collisions is identified.
• This net also possesses collision-free embeddings with the same symmetry, termed 'unforced' collisions.

Conclusions:

• The study demonstrates the diverse behaviors of unstable nets within crystallographic contexts.
• The concept of 'unforced' collisions provides a new perspective on net stability and symmetry.
• Findings contribute to a deeper understanding of network topology and symmetry in crystalline materials.