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Topology of Polymers

Koya Shimokawa, Kai Ishihara, and Yasuyuki Tezuka
Publication Date: 
Number of Pages: 
[Reviewed by
Greg McColm
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Every generation since the Victorian era must have seemed like the Age of Materials.  From Stanislas Sorel's artificial ivory through artificial dyes, artificial fibres, the plastic revolution, to ceramic superconductors, it's been one age of materials after another.  And that is unlikely to change, for many of our major technological challenges - from containing fusion to manipulating nanostructures - are problems in materials science.
By an interesting coincidence, this growth in materials science has tracked the development of the atomic theory and nanoscale descriptions of materials.  Many of these materials turn out to be polymers, which are described in the beginning of this book as “ubiquitous long chain molecules that are fundamental in biological as well as in industrial synthesis.”  The examples then trotted out - DNA, proteins, cellulosic compounds, fibers, films, plastics, and rubbers - are largely organic (this book steers clear of inorganic carbon examples like graphene and diamond), and largely concern discrete, if occasionally very large, molecules.
Molecules have been represented by graphs ever since August Hofmann used ball-and-stick models of molecules in his mid-Nineteenth century lectures, and like Hofmann, this book focuses on graphical representations of molecules sans geometric considerations.  The chemical families dominating this book are the alkanes and their more complex relatives.  An alkane can be regarded as a tree in which all non-leaf vertices (representing carbon atoms) are of degree four and the leaves represent hydrogen atoms.  A cyclo-alkane also consists of carbon and hydrogen atoms, although now the graphs can have cycles (but the nonleaf vertices are still of degree four, thus excluding chemicals like benzene).  Cyclo-alkanes have many graphical representations (and they are ubiquitous in modern society, especially as petrochemicals), and most examples in this book can be regarded as representations of cyclo-alkanes.
The graphs of alkanes and cyclo-alkanes are simplified by deleting all the leaves and then collapsing each maximal chain of degree 2 vertices into a single edge; the result is a multigraph whose vertices are all of degree 1, 3, or 4.  These multigraphs are not necessarily connected (but if disconnected, the components are inter-linked), and a graph of some number of carbon atoms,  connected in a cycle (with attending hydrogen leaves) is represented as a cycle consisting of one edge and no vertices.  One could say that much of the mathematical content of this book concerns multigraphs whose vertices are of degree 1, 3, or 4, and that many of the examples consist of numerous small 2-connected submultigraphs connected either by identifying a pair of vertices or by adding a bridging edge.
In a sequence of short chapters, each mathematically accessible and with numerous well-illustrated examples, the book introduces graphs, develops a scheme for classifying graphs representing alkanes and cyclo-alkanes, introduces some operations on these graphs, and reviews some elementary knot theory.  Then there is a chapter on the topological structure of molecules:  two alkanes may have the same number of carbon atoms and of hydrogen atoms (and thus be isomers), but they may be chemically distinct because their graphs are different.  The book concludes with a heavily chemical chapter on the synthesis and behavior of molecules of particular topologies.
This is a small book (81 pages) apparently aimed at a chemically sophisticated audience who were relative novices in combinatorics or topology.  Many chemical names flit about without being defined, but the book is still readable by novices in chemistry provided that the reader is  willing to treat these mysterious names as black boxes.
Greg McColm is a former mathematical logician who now works in geometric crystallography.   He is very behind on his Crystal Mathematician blog and needs to update his webpage.