New Developments in the Theory of Knots (Advanced Series in Mathematical Physics, Vol. 11)

Features

• Cover Type: Hard Cover with 800 pages
• Published by: World Scientific Pub Co Inc. December 1990
• Written in: English
• ISBN 10 Number: 9810201621
• ISBN 13 Number: 978-9810201623
• Book Dimensions: 9.9 x 6.8 x 2.1 inches
• Weighs: 3.7 pounds

Reader Reviews
This book is a collection of articles that are representative of the many exciting developments in knot theory that were occurring at the time of publication. Many of these developments have been extended and generalized since then, such as the theory of Vassiliev invariants, and thus the articles could be viewed as an introduction to this research. Hence it could still serve as a reference to the mathematical theory of knots and it relation to physics, via statistical mechanics and quantum field theory. I did not read all of the articles, so only a few of the ones I did will be reviewed here. The article by Vaughan Jones on polynomial invariants for knots via von Neumann algebras begins the collection and was definitely the tone-setting one of the time, due to the new invariants of knots discovered by Jones. The article discusses how to construct a polynomial invariant for tame oriented links using certain representations of the braid group. By using Markov's theorem and a trace on a type II(1) von Neumann algebra, the author shows that the invariant depends only on the closed braid. The von Neumann algebra is generated by an identity and a collection of projections, which satisfy certain types of relations. These relations involve a complex parameter, and when this parameter satisfies certain conditions there exists a trace on the von Neumann algebra which in turn satisfy a collection of relations. The relations on the projections and the trace determine the structure of the von Neumann algebra up to *-isomorphism. That the projection relations are similar to Artin's presentation of the braid group was what Jones and others to develop invariants of links and knots based on this trace. In another article Jones then obtains a polynomial invariant in two variables for oriented links that uses a trace on Hecke algebras "of type A", which was inspired by the connections with von Neumann algebras. His discussion in this article points out the need for a better understanding of the topological interpretation of these invariants. Pointing out that a more in-depth understanding of subfactors of finite index would assist in this topological interpretation, in a later article Jones outlines in more detail what is known for subfactors of finite index. The index, as defined by Jones, measures the size of a subfactor in a II(1) factor. In addition, Hans Wenzl discusses Hecke algebras of type A and subfactors, and shows how to compute the Jones index using AF algebras. The most provocative article in the book, and one not rigorous from a mathematical standpoint, is the article by Edward Witten on the quantum field theory and the Jones polynomial. The connection between these two seemingly disparate fields caused great excitiment in both the physics and mathematics communities, in spite of the fact that these results are unjustified mathematically, due to their reliance on path integrals. Witten was motivated in this article to find a three-dimensional interpretation of the Jones polynomial, which he does so via Yang-Mills theory in three dimensions. However, the Yang-Mills theory which he uses is not the standard one, but instead is based on the purely topological Chern-Simons theory. Witten considers the quantum field theory defined by the Chern-Simons theory and uses its gauge fields to define gauge-invariant observables. Because of the side-constraint of general covariance, these observables are chosen to be Wilson lines, which are independent of the metric. In an oriented three manifold Witten then considers oriented and non-intersecting knots and assigns a representation to each knots. Using the Chern-Simons three form Witten computes the path integral of the Wilson observables, and then proposes that these quantities are 3-dimensional interpretations of the Jones invariant. Witten first proves that the Chern-Simon form gives a meaningful quantum theory, i.e. that it is free from anomalies, and he justifies this by reducing the Chern-Simons invariant to a ratio of determinants, and then showing the absolute value of this ratio is the Ray-Singer analytic torsion. Witten then considers the calculation of the phase of the ratio, and then via the canonical quantization of the theory, shows how to obtain the desired knot invariants.