Knot theory now plays a large role in modern mathematics, and the most signifi-
significant results in this theory have been obtained in the last two decades. For scientific
research in this field, Jones, Witten, Drinfeld, and Kontsevich received the highest
mathematical award, the Fields medals. Even after these outstanding achievements,
new results were obtained and even new theories arose as ramifications of knot the-
theory. Here we mention Khovanov's categorification of the Jones polynomial, virtual
knot theory proposed by KaufTman and the theory of Legendrian knots.
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 fun-
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,
KaufTman, 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 rep-
representation and the newest A999—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.