The finding of invariants for knots has been a major unsolved problem in mathematics for over 125 years. The difficulty of the problem is attested to by the paucity of results over decades of time in the early 20th century. Then in the 1980's Vaughn Jones discovered some invariants that are related to ideas in physics, namely the theory of integrable models in statistical mechanics. This book is a collection of articles discussing the Jones work and other approaches that relate knot theory and statistical mechanics, written a few years after his discovery. My review will be confined to the articles which I read in detail.
An article by Vaughn Jones begins the book and discusses the connection between subfactors of von Neumann algebras and statistical mechanics. These von Neumann algebras occur as algebras of transfer matrices in statistical mechanics. These transfer matrices satisfy algebraic relations that are essentially the same as those appearing in special types of von Neumann algebras. The author makes it a point to discuss in detail the relevant constructions of von Neumann algebras, believing this has not been done in the literature. The von Neumann algebras related to the transfer matrices are particular types of II(1) and III factors, which the author constructs using Bratteli diagrams and the Gelfand-Naimark-Segal construction.
The article by Louis Kauffman discusses polynomial invariants of knots and the Yang-Baxter factorization equation. Polynomial invariants based on the Yang-Baxter equations are one-variable polynomials. The author points out that it is unknown whether two-variable invariants can be extracted from the Yang-Baxter equations, but points out how to construct these using skein models.
The article by Michio Jimbo is an introduction to the Yang-Baxter equation with emphasis on the role of quantum groups. The solutions of the Yang-Baxter equation are discussed in the light of the work of A. Belavin and V.G. Drinfeld in the context of simple Lie algebras. The author shows in this case that the solutions are either elliptic, trigonometric, or rational functions. This is followed by a discussion of how to "quantize" this situation, which leads to the theory of quantum groups, a field that has grown considerably since this article was written. The author discusses a particular example of a quantum group, called the universal enveloping algebra, and studies its representations and the Drinfeld universal R matrix. The "classical" Yang-Baxter r-matrix is then the classical limit of this R-matrix. The author shows how to obtain higher representations by using an analog of the technique of constructing irreducible representations of Lie algebras by forming tensor products of fundamental representations and decomposing them. This technique is known as the fusion procedure here and elsewhere in the literature.
The article by Toshitake Kohno is a review article on representations of the braid group with respect to the Yang-Baxter equation for face models in statistical mechanics. The representations of the braid group appear explicitly as the monodromy of integrable connections defined for any simple Lie algebra and its irreducible representation. Interestingly, the connections describe n-point functions in a conformal field theory on the Riemann sphere with gauge symmetry. The author begins with a finite-dimensional complex simple Lie algebra and its irreducible representation. Selecting an orthonormal basis of this Lie algebra with respect to the Cartan-Killing form, the author constructs certain matrices in the endomorphisms of the n-fold tensor product of the representation. These matrices satisfy certain relations that are a special case of the Yang-Baxter equation. A connection defined using these matrices and a complex parameter ranging over a set consisting complex n-vectors with unequal coordinates is shown to be integrable using these relations. The fundamental group of the complex parameter set is the 'pure braid group with n strings' and the (quadratic) relations are viewed as an infinitesimal version of the defining relations of the pure braid group. By taking the quotient of the parameter set with the symmetric group one obtains the braid group, the representations of which are consequently obtained using the monodromy of this connection. Another representation is derived from the quantized universal enveloping algebra of the Lie algebra.
By far the most interesting article, and the one least rigorous mathematically, is the one by Edward Witten on quantum field theory and the Jones polynomial. The author shows that a Yang-Mills theory in 2 + 1 dimensions consisting of merely the Chern-Simons terms is exactly soluble and can be used to give the Jones polynomial a three-dimensional interpretation, which was highly desired at the time of writing. He also shows that the Jones polynomial can be generalized from the 3-sphere to arbitrary 3-manifolds, and gives invariants for these manifolds, which can be computed from a surgery presentation. The author's constructions are fascinating, particularly from a physics standpoint, but mathematically they are very suspect, since they are dependent on the notion of a path integral. The latter, despite decades of concentrated effort, has defied a mathematically rigorous formulation. The results in the article have thus been classified as "physical mathematics", and therefore conjectural and tentative from a purely mathematical standpoint. This is a fair classification, and it motivated other mathematicians to find alternative formulations that are well-defined mathematically. Indeed, this article has resulted in an explosion of research on both knot invariants and invariants for 3-manifolds, some of which has remain tied to quantum field theory, and some making a concentrated effort to remove these invariants from their dependence on it.