Knot theory is one of those things in contemporary mathematics which seems to have it all. First and foremost, it is of course serious mathematics. But it has so many other things going for it. It is seductive due to its objects of study being visualisable (at least for the less rococo knots in this game: any one can draw a trefoil and is easily led to believe he can draw nastier knots, too — and sometimes this is true). It is connected to all sorts of wonderful other mathematics. And it has begun to exhibit a dramatic amenability to applications to the sciences and even to engineering, to the extent of attracting even distant outsiders. Additionally, as far as visualizing what goes on in the nastier parts of knot theory is concerned, the revolution in computer graphics has of course opened all sorts of doors to perception (with apologies to Aldous Huxley), and in point of fact the book under review is a fine example of this feature: the artwork is wonderful, and this holds also regarding the book’s more prominent focus, viz. another element of low-dimensional topology: embedded graphs.
It is actually the case that knot theory per se only makes it appearance toward the end of the book, in the fifth chapter, while the preceding four chapters are concerned with embedded graphs, dualities, and graph polynomials. But this all fits with the authors’ goals which explicitly include to “[illustrate] the interdependency between duality, medial graphs, and knots; how this interdependency is reflected in algebraic invariants of graphs and knots; and how this interdependency can be exploited to solve problems in graph theory and knot theory.”
Thus, Graphs on Surfaces: Dualities, Polynomials, and Knots deals integrally with another mathematical discipline which, however, has not (yet?) enjoyed the quite the same level of air-play that knot theory has begun to enjoy: graph theory as a part of, in the broad sense, combinatorics. To be sure, what with the advent of ever-more-sophisticated computing, graph theory has also begun to undergo dramatic evolution, but it does not yet sport the same extramural connections that knot theory has. This said, however, it is worth noting that (as per their university web-pages) the authors include in their interests such things as graph theory in the design of computer chips and in biology, so the game is certainly afoot.
In any case, both of the book’s foci, graphs and knots, are occasions for very nice mathematics, and the authors have composed a very interesting and valuable work. It’s not for raw beginners: the authors recommend studying Topological Graph Theory by Tucker and Gross, or Graphs on Surfaces by Thomassen and Mohar as a prerequisite, and it is clearly indicated that the reader should have some knowledge of knot theory, too — in this area the choices are many and varied, of course, but here are a quasi-random three: Knot Theory by Livingston (an MAA book), An Introduction to Knot Theory by Lickorish, and Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-dimensional Topology by Prasolov.
For properly prepared readers, however, the book under review is the occasion for all sorts of fun including the inner life of ribbon groups, Tait graphs, Penrose polynomials, Tutte polynomials, and of course Jones polynomials and HOMFLY polynomials. This is fascinating mathematics, presented in a clear and accessible way.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
1. Embedded Graphs
2. Generalised Dualities
3. Twisted duality, cycle family graphs, and embedded graph equivalence
4. Interactions with Graph Polynomials
5. Applications to Knot Theory