By K. A. Broughan

**Read Online or Download Invariants for Real-Generated Uniform Topological and Algebraic Categories PDF**

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During this quantity, that's devoted to H. Seifert, are papers in line with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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Thus, the Petersen graph is the smallest projective fullerene. In general, the projective fullerenes are exactly the antipodal quotients of the centrally symmetric spherical fullerenes. Thus, the problem of enumeration and construction of projective fullerenes reduces simply to that for centrally symmetric conventional spherical fullerenes. The point symmetry groups that contain the inversion operation are Ci , Cmh , (m even), Dmh (m even), Dmd (m odd), Th , Oh , and Ih . A spherical fullerene may belong to one of 28 point groups ([FoMa95]) of which eight appear in the previous list: Ci , C2h , D2h , D6h , D3d , D5d , Th , and Ih .

Spherical and toroidal fullerenes have an extensive chemical literature, and Klein bottle polyhexes have been considered, for example, in [Kir97, KlZh97]. Note that at least one spherical fullerene with v vertices exists for all even v with v ≥ 20, except for the case v = 22 ([GrMo63]). 2 Toroidal and Klein bottle fullerenes The (6, 3)-tori and (6, 3)-Klein bottles are related to {6, 3} in a straightforward way. The underlying surfaces are quotients of the Euclidean plane R2 under groups of isometries generated by two translations (for T2 ) or one translation and one glide reflection (for K2 ).

So, F and F intersect in an edge, vertex, or ∅. The same proof works for other intersections of cells. 3 If a finite (r, q)-polycycle has a boundary vertex whose degree is less than q, then the total number of these vertices is at least two. Proof. Take a flag f in this (r, q)-polycycle P and a flag f in {r, q}. 1. The boundary B of P consists of vertices v1 , . . , vk . The image of this boundary in {r, q} is also a cycle φ(B). 50 Chemical Graphs, Polycycles, and Two-faced Maps Whenever B has a vertex vi of degree q, the edges ei = vi−1 vi and ei+1 = vi vi+1 belong to a common face F.