When Topology Meets Chemistry

Abstract

The applications of topological techniques for understanding molecular structures have become increasingly important over the past thirty years. In this topology text, the reader will learn about knot theory, 3-dimensional manifolds, and the topology of embedded graphs, while learning the role these play in understanding molecular structures. Most of the results that are described in the text are motivated by questions asked by chemists or molecular biologists, though the results themselves often go beyond answering the original question asked. There is no specific mathematical or chemical prerequisite; all the relevant background is provided. The text is enhanced by nearly 200 illustrations and more than 100 exercises. Reading this fascinating book, undergraduate mathematics students can escape the world of pure abstract theory and enter that of real molecules, while chemists and biologists will find simple, clear but rigorous definitions of mathematical concepts they handle intuitively in their work.

Toroidal embeddings of abstractly planar graphs are knotted or linked

“…Step (7) shows that each spatial fragment can individually ambient isotoped to a planar embedded spatial fragment in space, independently from all other spatial fragments. The combination of the ambient isotopies given in step (3), step (6) and step (7) gives the desired deformation of G into the plane which proves Theorem 1.…”

“…The interaction between mathematical topology, in particular topological graph theory, and the investigation of chemical structures is a rich area ( [1][2][3][4][5][6][7] and the references therein). Topological graph theory studies embeddings of graphs in 3-space.…”

“…Results about spatial graphs directly translate to information about the configuration of molecules. In particular, if entangled chemical structures like knots, links, braids and ravels are present, topological graph theory can be an appropriate framework [1][2][3][4][5][6][7].…”

Surmounting the Cartesian Cut: Klein Bottle Logophysics, The Dirac Algebra and the Genetic Code

“…This construction does not require the assumption of the double helix (DH); the latter is bound to one single subalphabet which is already evident in the No.3 reading of P(n) which instead yields a MB.A MB model for circular genetic code was proposed (Burdick,1971), yet to our knowledge, this has not been the case for a KB structure (Bonchev and Rouvray, 2000;Flapan, 2000). These findings, which in our present discussion has neglected the topology of the p-orbitals of aromatics crucial to DNA, support the unifying paradigm for chemistry (Bonchev and Rouvray, 2000;Herges, 2006;Sokolov 1979) that claims that the topology of molecules, rather than their hybridization geometry relations, characterizes their stereochemical configuration, which we have suggested to be crucial to allosterics, cell biology and embryological differentiation (Rapoport, to appear & 2011c).…”

Topological Symmetry Groups of Small Complete Graphs

“…Because of these rotating pieces, this molecule is achiral though it cannot be rigidly superimposed on its mirror form. A detailed discussion of the achirality of this molecule can be found in [1]. Figure 1.…”