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Introduction to Vassiliev knot invariants. (English) Zbl 1245.57003

Cambridge: Cambridge University Press (ISBN 978-1-107-02083-2/hbk; 978-1-139-41581-1/ebook). xvi, 504 p. (2012).
A major goal of knot theory is to distinguish knots. Numerous knot invariants have been constructed for this purpose. Some of these come from algebraic topology, some come from geometry, and some even come from operator algebras. As if we didn’t have enough knot invariants already, Victor Vassiliev, as a byproduct of his research on singularity theory, uncovered a new infinite family of knot invariants in the late 1980’s. These same knot invariants were discovered independently by Goussarov. Vassiliev invariants have not enabled us to distinguish knots which we did not know how to distinguish before, nor have they led to proofs of outstanding topological conjectures. Their interest rather is in that they reveal a previously unknown and unexpected local (or quantum) nature to knots – that substantial information about a knot can be gleaned from knowing the knot in a few small neighbourhoods, together with combinatorial information about how strands in these neighbourhoods connect to one another. Another source of interest in Vassiliev invariants is that they form a graded algebra whose properties mimic the properties of a graded algebra of polynomials. This stems from the similarity between the Vassiliev skein relation used to define Vassiliev invariants and FrĂ©chet’s additive characterization of polynomials. So the study of Vassiliev knot invariants may roughly be thought of as the study of polynomial functions from a space of knots to the base ring. Moreover, a ‘Taylor Theorem’ is conjectured to hold, in the sense that any knot invariant is conjectured to be approximated by a series of Vassiliev invariants. Thus, Vassiliev invariants emerge as a potential organizing and unifying concept in knot theory.
The book under review, written by three leading researchers in the field, focuses on Vassiliev knot invariants, stressing combinatorial aspects. It is the first textbook on this much-studied topic, filling a gap in the literature.
The book may roughly be divided into four parts. Chapters 1–4 introduce knots, knot polynomials, and finite-type invariants. Chapters 5–7 study the graded Hopf algebra of Jacobi diagrams and Lie algebra weight systems. The highlight of the book for the reviewer comes in Chapters 8–10, which provide a very nice exposition of the Kontsevich integral. Finally, Chapters 11–15 consist of a hodgepodge of odds and ends related to Vassiliev knot invariants.
It is clear that this book is a labour of love, and that no effort has been spared in making it a useful textbook and reference for those seeking to understand its subject. The target readership consists both of first-time learners, for whom pedagogically sound explanations and numerous well-chosen exercises are provided to enhance comprehension, and of experienced mathematicians, for whom many tables of data and readable concise guides to research literature are provided. Numerous figures are included to supplement written explanations. The content is well-modularized, in the sense that different sections of the book may be read independently of one another, and that when there is an essential dependence between sections then this fact is clearly pointed out and the relationship between the sections is explained. This, and a thorough index, combine to make this book not only a valuable textbook, but also a valuable reference.

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M30 Wild embeddings
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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