Zigzags, Railroads, and Knots in Fullerenes M. Deza and M. Dutour CNRS and LIGA, E Ä cole Normale Supe ´rieure, 45 rue d’Ulm, 75230 Paris, France P. W. Fowler* Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, U.K. Received January 29, 2004 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 e 74) examples are found for C 34 and all C n with n g 38, all chiral, belonging to groups C 1 , C 2 , C 3 , D 3 , or D 5 . An r-knot fullerene has a railroad corresponding to the projection of a nontrivial knot: examples are found for C 52 (trefoil), C 54 (figure-of-eight or Flemish knot), and, with isolated pentagons, at C 96 ,C 104 ,C 108 ,C 112 ,C 114 . 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 Kekule ´ 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. 1. INTRODUCTION Two structural vectors which play important roles in the classification of polyhedra are the vertex-degree vector V) {V 3 , V 4 , ...} and the face-size vector p ) {p 3 , p 4 , ...}, where V r and p r are the numbers of vertices of degree r and faces of size r, respectively. Euler’s theorem gives necessary conditions 1 for realizability of a given V, p pair (the Eberhard problem 2 ). Entries in V and p satisfy Sufficient conditions are known for various classes of polyhedra. 2-4 Alternative formulations of the third require- ment are sometimes seen. 5 Some polyhedra of particular interest in carbon chemistry are the fullerenes, C n , which are trivalent (V 3 ) n, V r ) 0, otherwise), have only pentagonal and hexagonal faces (p 5 ) 12, p 6 ) n/2 - 10, p r ) 0, otherwise), and are mathematically realizable for all n ) 20 + 2p 6 with the exception of p 6 ) 1. 4 Isomeric fullerenes share V- and p-vectors but differ in the arrangement of their 12 pentagonal faces. A number of fullerene polyhedra with vertex counts in the range of 60 to ∼100, all with disjoint pentagonal faces (isolated-pentagon fullerenes, in the chemical terminology), have been characterized as all-carbon molecular cages. When n is of moderate size, a useful notation, taken up in IUPAC nomenclature, is to label isomers of the C n fullerene n:m where m is the place of the isomer in the lexicographical order of spiral codes for the set of general or isolated- pentagon fullerenes. 6 The present paper is concerned with a third structural property, the zigzag vector z, which carries information about the edges of the polyhedron and makes finer distinctions within sets of isomeric polyhedra. This vector can provide an overwhelming amount of detail for general polyhedra, and we intend here only to identify some basic notions and use them as a framework for the study of the zigzag vectors of fullerenes. In particular, we discuss two connections between fullerene zigzag vectors and knots. Knots appear for fullerenes in at least two ways. First, in what may be considered the simplest case of zigzag structure, when a given fullerene has only a single zigzag circuit, the Schlegel diagram of the fullerene yields a projection of a knot. It will be shown that some (minimal) knots then define a unique Kekule ´ structure (perfect matching) on the corre- sponding fullerene, thus establishing an unexpected connec- tion between mathematical knots and the electronic structure of fullerenes as chemical entities. Second, among fullerenes with more than one zigzag circuit, some exhibit a pairing of zigzag circuits in which two parallel (i.e. nonintersecting, concentric) circuits of edges bound a circuit of hexagonal faces, or “simple railroad”. The concept can be extended to include self-intersecting railroads. Simple and (doubly) self-intersecting railroads are associated with projections of alternating knots defined by the adjacencies of the hexagonal faces. Simple railroads give a point of departure for cylindrical expansion of fullerenes and a formal route to construction of capped nanotubes. * Corresponding author phone: 44 1392 263 466; fax: 44 1392 263 434; e-mail: P.W. Fowler@ex.ac.uk. ∑ r (6 - r)V r g 12; ∑ r (6 - r)p r g 12; ∑ r (4 - r)(V r + p r ) ) 8 (1) 1282 J. Chem. Inf. Comput. Sci. 2004, 44, 1282-1293 10.1021/ci049955h CCC: $27.50 © 2004 American Chemical Society Published on Web 05/12/2004