Cells complexes

Jourjine [jour85] generalizes Euclidean lattice field theory on a d-dimensional lattice to a cell complex. He uses homology theory to replace points by cells of various dimensions and fields by functions on cells, the cochains, in hopes of developing a formalism that treats space-time as a dynamical variable and describes the change in the dimension of space-time as a phase transition (see figure 12.19). 

The cell surface contains antigens, which are referred to as CD, which stands for cluster of differentiation. The antibodies are produced against a specific antigen. When administered, usually by an intravenous injection, the antibody binds to the antigen, which may trigger the immune system to result in cell death through complement-mediated cellular toxicity, or the antigen-antibody cell complex may be internalized to the cancer cell, which results in cell death. Monoclonal antibodies also may carry radioactivity, sometimes referred to as hot antibodies, and may be referred to as radioimmunotherapy, so the radioactivity is delivered to the cancer cell. Antibodies that contain no radioactivity are referred to as cold antibodies. 

Lipids A first induce the expression of early inflammatory genes such as tumor necrosis factor- (TNF)-a, EL-1(3, type 2 TNF receptor, IP-10, D3, D8 and D2 genes [34]. Then further genes are activated such as other cytokines and receptors, adhesion molecules, acute-phase proteins, tissue factors, as well as the inducible NOS (NOS II). These cascades of events initiated by lipid A provoke in their target cells complex responses in vivo, whose relevance in the host response to tumor growth is reviewed below. 

We shall now see how to apply the theorem to the molecular trefoil knot, which was illustrated in Figure 17. We can create a molecular cell complex G by replacing each isolated benzene ring by a cell and each chain of three fused rings by a single cell. We prove by contradiction that our molecular cell complex is topologically chiral. Suppose that it is topologically achiral. Then there is a defor-... 

In a similar way we can prove that the embedded cell complex of the molecular (4,2)-torus link (see Figure 18) is topologically chiral. Also, by adding appropriate labels we can similarly prove the topological chirality of the oriented embedded cell complex of the molecular Hopf link (see Figure 19). 

Liang and Mislow s idea of contracting naphthalenes can be formalized if we consider the molecular cell complex rather than the molecular graph of triple layered naphthalenophane. We obtain a cell complex G by replacing each naphthalene by a pair of cells. We shall now prove by contradiction that this mo-... 

Figure 29. The cell complex of triple-layered naphthalenophane together with the arcs ax, a2, and a3.

By using a similar method, we can prove that the cell complexes of the molecules [m [n]paracyclophane [22] and triple layered cyclophane [23] are also intrinsically chiral. Figure 30 illustrates these molecules... 

A closed map, cell-complex of a polyhedron. It is a 5Rq, 4/ 2 plane graph (see Chapter 9)... 

If M is a closed map, then we can define its dual map M by interchanging faces and vertices. See Section 4.1 for some related duality notions for non-closed maps. A map is called a cell-complex if the intersection of any two faces, edges, or vertices... 

A map M is called reduced (see [Moh97, Section 3]) if its universal cover is 3-connected and is a cell-complex. It is shown in [Moh97, Corollary 5.4] that reduced maps admit unique primal-dual circle packing representations on a Riemann surface of the same genus moreover, a polynomial time algorithm allows one to find the coordinates of those points relatively easily. This means that the combinatorics of the map determines the structure of the Riemann surface. 

Denote by Bundlem, m > 2, the plane graph with two vertices and m edges between them (so, m 2-gonal faces). The plane graph Bundlem, which is dual to m-gon, has the symmetry group Dmh = 7 (2,2, m) and it is a regular map, which is not a cell-complex. 

At least one toroidal polyhex that is cell-complex exists for all numbers of vertices v > 14. The unique cell-complex toroidal fullerene at v = 14 is a realization of the Heawood graph. It is GC2,i(hexagon) in terms of Goldberg-Coxeter construction and is the dual of K7, which itself realizes the 7-color map on the torus. This map and its dual are shown in Figure 3.1. 

In this terminology, our definition of projective fullerenes amounts to selection of cell-complex projective-planar 3-valent maps with only 5- and 6-gonal feces. As noted above, P5 — 6 for these maps. Thus, the Petersen graph is die smallest projective fullerene. In general, the projective fullerenes are exactly the antipodal quotients of the centrally symmetric spherical fullerenes. 

We will prove later (in Theorem 4.3.2) that all vertices, edges, and interior faces of an (r, )-polycycle form a cell-complex (see Section 1.2.1). 

Proof, To prove that it is a cell-complex, we shall prove that the intersection of any two cells (i.e. vertices, edges, or interior faces) of an (r, )-polycycle P is again a cell of P or 0. For the intersection of vertices with edges or faces this is trivial. For the intersection of edges or faces, we will use the cell-homomorphism

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