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Minimal-transitivity finite tangles and catenated polyhedra
Michael O'Keeffe1, Michael M J Treacy2
1School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, USA.
Acta Crystallographica. Section A, Foundations and Advances
|January 22, 2026
Summary
This study explores graph embeddings called tangles, which include knot and link substructures. Researchers identified eight families of catenated polyhedra and ten families of tangles with specific symmetry properties.
Area of Science:
- Knot theory
- Graph theory
- Computational topology
Background:
- Tangles are graph embeddings containing knots or links.
- Understanding tangle structures is crucial in topology and graph theory.
Purpose of the Study:
- To enumerate and describe piecewise-linear embeddings of cubic and icosahedral tangles.
- To classify linked polyhedra with specific transitivity properties.
Main Methods:
- Enumeration of tangle and linked polyhedron families.
- Description of embeddings based on symmetry and edge types.
- Focus on members with the largest girth.
Main Results:
- Identified eight families of catenated polyhedra.
- Identified ten families of tangles.
- Characterized members with the largest girth within each family.
Conclusions:
- Provides a systematic classification of specific tangle and linked polyhedron families.
- Establishes a framework for studying complex topological structures in graphs.
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