Zigzags, Railroads, and Knots in Fullerenes

Two connections between fullerene structures and alternating knots are established. Knots may appear in two ways:  from zigzags, i.e., circuits (possibly self-intersecting) of edges running alternately left and right at successive vertices, and from railroads, i.e., circuits (possibly self-intersecting) of edge-sharing hexagonal faces, such that the shared edges occur in opposite pairs. A z-knot fullerene has only a single zigzag, doubly covering all edges:  in the range investigated (n ≤ 74) examples are found for C₃₄ and all C_n with n ≥ 38, all chiral, belonging to groups C₁, C₂, C₃, D₃, or D₅. An r-knot fullerene has a railroad corresponding to the projection of a nontrivial knot:  examples are found for C₅₂ (trefoil), C₅₄ (figure-of-eight or Flemish knot), and, with isolated pentagons, at C₉₆, C₁₀₄, C₁₀₈, C₁₁₂, C₁₁₄. Statistics on the occurrence of z-knots and of z-vectors of various kinds, z-uniform, z-transitive, and z-balanced, are presented for trivalent polyhedra, general fullerenes, and isolated-pentagon fullerenes, along with examples with self-intersecting railroads and r-knots. In a subset of z-knot fullerenes, so-called minimal knots, the unique zigzag defines a specific Kekulé structure in which double bonds lie on lines of longitude and single bonds on lines of latitude of the approximate sphere defined by the polyhedron vertices.