Knot Theory

Rating:
Summary: Best of Knots
Review: Knot Theory by Vassily Manturov (CRC Press) The aim of the present monograph is to describe the main concepts of modern knot theory together with full proofs that would be both accessible to beginners and useful for professionals. Thus, in the first chapter of the second part of the book (concerning braids) we start from the very beginning and in the same chapter construct the Jones two-variable polynomial and the faithful representation of the braid groups. A large part of the present title is devoted to rapidly developing areas of modern knot theory, such as virtual knot theory and Legendrian knot theory.
In the present book, we give both the "old" theory of knots, such as the fundamental group, Alexander's polynomials, the results of Dehn, Seifert, Burau, and Artin, and the newest investigations in this field due to Conway, Matveev, Jones, Kauffman, Vassiliev, Kontsevich, Bar-Natan and Birman. We also include the most significant results from braid theory, such as the full proof of Markov's theorem, Alexander's and Vogel's algorithms, Dehornoy algorithm for braid recognition, etc. We also describe various representations of braid groups, e.g., the famous Burau representation and the newest (1999-2000) faithful Krammer-Bigelow representation. Furthermore, we give a description of braid groups in different spaces and simple newest recognition algorithms for these groups. We also describe the construction of the Jones two-variable polynomial.
In addition, we pay attention to the theory of coding of knots by d-diagrams, described in the author's papers. Also, we give an introduction to virtual knot theory, proposed recently by Louis H. Kauffman. A great part of the book is devoted to the author's results in the theory of virtual knots.
Proofs of theorems involve some constructions from other theories, which have their own interest, i.e., quandle, product integral, Hecke algebras, connection theory and the Knizhnik-Zamolodchikov equation, Hopf algebras and quantum groups, Yang-Baxter equations, LD-systems, etc.
The contents of the book are not covered by existing monographs on knot theory; the present book has been taken a much of the author's Russian lecture notes book on the subject. The latter describes the lecture course that has been delivered by the author since 1999 for undergraduate students, graduate students, and professors of the Moscow State University.
The present monograph contains many new subjects (classical and modern), which are not represented in the author's earlier Russian version of this book.
While describing the skein polynomials we have added the Przytycky-Traczyk approach and Conway algebra. We have also added the complete knot invariant, the distributive grouppoid, also known as a quandle, and its generalisation. We have rewritten the virtual knot and link theory chapter. We have added some recent author's achievements on knots, braids, and virtual braids. We also describe the Khovanov categorification of the Jones polynomial, the Jones two-variable polynomial via Hecke algebras, the Krammer-Bigelow representation, etc.
The book is divided into thematic parts. The first part describes the state of "pre-Vassiliev" knot theory. It contains the simplest invariants and tricks with knots and braids, the fundamental group, the knot quandle, known skein polynomials, Kauffman's two-variable polynomial, some pretty properties of the Jones polynomial together with the famous Kauffman-Murasugi theorem and a knot table.
The second part discusses braid theory, including Alexander's and Vogel's algorithms, Dehornoy's algorithm, Markov's theorem, Yang-Baxter equations, Burau representation and the faithful Krammer-Bigelow representation. In addition, braids in different spaces are described, and simple word recognition algorithms for these groups are presented. We would like to point out that the first chapter of the second part (Chapter 8) is central to this part. This gives a representation of the braid theory in total: from various definitions of the braid group to the milestones in modern knot and braid theory, such as the Jones polynomial constructed via Hecke algebras and the faithfulness of the Krammer-Bigelow representation.
The third part is devoted to the Vassiliev knot invariants. We give all definitions, prove that Vassiliev invariants are stronger than all polynomial invariants, study structures of the chord diagram and Feynman diagram algebras, and finally present the full proof of the invariance for Kontsevich's integral. Here we also present a sketchy introduction to Bar-Natan's theory on Lie algebra representations and knots. We also give estimates of the dimension growth for the chord diagram algebra.
In the fourth part we describe a new way for encoding knots by d-diagrams proposed by the author. This way allows us to encode topological objects (such as knot, links, and chord diagrams) by words in a finite alphabet. Some applications of d-diagrams (the author's proof of the Kauffman-Murasugi theorem, chord diagram realisability recognition, etc.) are also described.
The fifth part contains virtual knot theory together with "virtualisations" of knot invariants. We describe Kauffman's results (basic definitions, foundation of the theory, Jones and Kauffman polynomials, quandles, finite-type invariants), and the work of Vershinin (virtual braids and their representation). We also included our own results concerning new invariants of virtual knots: those coming from the "virtual quandle", matrix formulae and invariant polynomials in one and several variables, generalisation of the Jones polynomials via curves in 2-surfaces, "long virtual link" invariants, and virtual braids.
The final part gives a sketchy introduction to two theories: knots in 3-manifolds (e.g., knots in RP3 with Drobotukhina's generalisation of the Jones polynomial), the introduction to Kirby's calculus and Witten's theory, and Legendrian knots and links after Fuchs and Tabachnikov. We recommend the newest book on 3-manifolds by Matveev.
At the end of the book, a list of unsolved problems in knot and link theory and the knot table are given.
The description of the mathematical material is sufficiently closed; the mono-graph is quite accessible for undergraduate students of younger courses, thus it can be used as a course book on knots. The book can also be useful for professionals because it contains the newest and the most significant scientific developments in knot theory. Some technical details of proofs, which are not used in the sequel, are either omitted or printed in small type.