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Tangled (up in) cubes.

S T Hyde¹, G E Schröder-Turk

¹Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 0200, Australia. stephen.hyde@anu.edu.au

Acta Crystallographica. Section A, Foundations of Crystallography

|February 16, 2007

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Researchers enumerated simple cube graph entanglements, forming hexagonal mesh complexes on a torus. Five chiral pairs of knotted graphs were discovered, including torus knots and links.

Area of Science:

Mathematics
Graph Theory
Topology

Background:

Knot theory and graph theory explore complex structures.
Torus knots and links are fundamental objects in topology.
Cube graphs provide a basis for studying entanglements.

Purpose of the Study:

To enumerate the simplest entanglements of the cube graph's edges.
To construct and analyze two-cell {6, 3} complexes on a genus-one torus.
To identify and classify chiral knotted and linked subgraphs.

Main Methods:

Enumeration of graph entanglements.
Construction of {6, 3} hexagonal mesh complexes on a torus.
Identification of knotted and linked subgraphs using topological invariants.

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Main Results:

Successfully enumerated the simplest entanglements of the cube graph.
Formed two-cell {6, 3} complexes on a genus-one torus.
Discovered five chiral pairs of knotted graphs, including (2,2) and (2,4) torus links and (3,2) and (4,3) torus knots.

Conclusions:

The study provides a foundational enumeration of simple cube graph entanglements.
The identified complexes and chiral pairs offer new examples in topological graph theory.
This work contributes to the understanding of knots and links in complex structures.