Zigzags, railroads, and knots in fullerenes

Area of Science:

• Mathematical Chemistry
• Computational Chemistry
• Graph Theory

Background:

• Fullerenes are allotropes of carbon with spherical or ellipsoidal molecules.
• Knot theory is a branch of topology studying mathematical knots.
• Previous research has explored geometric and topological properties of fullerenes.

Purpose of the Study:

• To establish connections between fullerene structures and alternating knots.
• To investigate two distinct mechanisms for knot formation in fullerenes: zigzags and railroads.
• To analyze the occurrence and properties of these knot structures in various fullerene types.

Main Methods:

• Identification of knot structures based on edge circuits (zigzags) and face circuits (railroads).
• Analysis of fullerene structures, including trivalent polyhedra and isolated-pentagon fullerenes.
• Statistical analysis of knot occurrences and classifications (z-knot, r-knot, z-vectors).

Main Results:

• Two types of fullerene knots identified: z-knots (single zigzag) and r-knots (railroad projection of nontrivial knots).
• Examples of z-knot fullerenes found for C34 and C(n) with n ≥ 38.
• Examples of r-knot fullerenes identified, including trefoil and figure-of-eight knots in specific fullerene structures (e.g., C52, C54, C96).

Conclusions:

• Fullerene structures exhibit inherent knotting properties through specific geometric configurations.
• The study provides a framework for classifying and understanding knot types within fullerene chemistry.
• Minimal knots reveal a specific Kekulé structure where double bonds align with longitudinal lines.