Introduction to Vassiliev Knot Invariants

This book is a detailed introduction to the theory of finite type (Vassiliev) knot invariants, with a stress on its combinatorial aspects. It is intended to serve both as a textbook for readers with no or little background in this area, and as a guide to some of the more advanced material. Our aim is to lead the reader to understanding by means of pictures and calculations, and for this reason we often prefer to convey the idea of the proof on an instructive example rather than give a complete argument. While we have made an effort to make the text reasonably self-contained, an advanced reader is sometimes referred to the original papers for the technical details of the proofs. Version 3: some typos and inaccuracies are corrected.Comment: 512 pages, thousands picture

The Kontsevich integral

The paper contains a detailed exposition of the construction and properties of the Kontsevich integral invariant, crucial in the study of Vassiliev knot invariants. Mathematics Subject Classification (1991): 57M52. Keywords: Vassiliev invariants, Kontsevich integral. This expository paper grew out of several lectures given by the authors in different places (Moscow and Pereslavl-Zalessky, Russia; Pisa, Italy; Bath, UK; Aizu-Wakamatsu, Japan; Utrecht, Netherlands). Its purpose is to explain the famous contribution of Maxim Kontsevich to Vassiliev knot invariant theory. More specifically, our aim is to present a proof of Kontsevich's fundamental theorem, which states that for each weight system there exists a Vassiliev invariant whose symbol is exactly the given weight system, or, in other words, that the only relations in the graded algebra of Vassiliev knot invariants are the 1- and 4-term relations. Recently, several papers dedicated to the Kontsevich integral have appeared: see ..