Knot theory, or, more properly, knot and link theory, is very hot stuff in today’s mathematics, and for many good reasons. Possibly the most evocative one, for me at least (as an outsider with a somewhat parochial view), is the fact that over the last thirty years or so the spectacular growth of low-dimensional topology in conjunction, or maybe in collusion, with quantum physics, provides one of the truly grand examples of synergy in the history of mathematics (as well as physics). Quantum field theory, in particular, has provided magnificent results in this regard, starting with Witten’s approach to the Jones polynomial by means of gauge theory, specifically Chern-Simons theory. And this was just the opening salvo: we now have a plethora of knot invariants to play with, including, for example, the HOMFLY polynomial, Khovanov homology, and the subject of the present book, Vassiliev’s invariants.

The object of the game is to determine (isotopy) invariants of knots, i.e. mathematical objects one attaches to knots which stay put as the knot is deformed in reasonable topological ways. (Here we encounter the yoga of Reidemeister moves: they retain the knot’s knottedness without making things worse.) The ideal invariant, one hopes, will tag the knot uniquely, but that is really a rather difficult thing to achieve: the gold-standard of knot invariants, for instance, the famous Jones polynomial, is not known even to be able to detect the unknot, whereas some homologies attached to knots do achieve this. For example, both instanton knot Floer homology and Khovanov homology detect the unknot: there is the striking result due to Peter Kronheimer and Tomasz Mrowka to the effect that a knot is the unknot (i.e. not a knot at all, just a simple loop modulo the obvious topological stuff) if and only if the attached reduced Khovanov homology has rank one. By the way the aforementioned duo prove this result by constructing a spectral sequence starting with (reduced) Khovanov homology and abutting to an already known unknot detector, instanton knot Floer homology. Clearly this is pretty sexy stuff.

So, in any case, there’s quite a menagerie of knot invariants, and one really does need pretty good road-maps to navigate through the forest they inhabit. A great guide indeed, and probably *the* place to start (modulo the usual condition of mathematical sophistication) is the 1993 Bulletin AMS article, “New points of view in knot theory,” by Joan Birman: its introduction affords us the right perspective on at least some of the players mentioned above. Birman starts off by noting that “every generalized Jones invariant [certainly including the Jones polynomial itself is included in this mix] is obtained from a trace function on an ‘R-matrix representation’ of a family of braid groups [coming from] the theory of quantum groups. For this reason, the collection of knot and link invariants that they determine have been called quantum group invariants.” She goes on quickly to present a very beautiful characterization of Vassiliev invariants in close relation to these quantum group invariants as follows: Let \(J(K,q)\) be a quantum group invariant for a knot \(K\) and consider the power series \(P(K,x) = \sum_{i\geq 0} u_i(K)x^i\), with rational coefficients, which is obtained by substituting \(1=e^x\)in \(J(K,q)\) and expanding the powers of \(e^x\) that occur into Taylor series. Then \(u_i(K)\) is 1 for \(i=0\) and is an order \(i\) Vassiliev invariant if \(i\geq 1\) Again, extremely sexy stuff.

So, we may certainly stipulate without fear of dispute that Vassiliev invariants are of huge interest to knot theory in general, and worthy of very serious study by any low-dimensional topologist or knot theorist (or even a fellow traveler). The book by Chmutov, Duzhin, and Mostovoy, under review, is a high-level introduction to this material. We learn early on that Victor Vassiliev was working on determinants in spaces of smooth maps when he discovered his invariants, and it was none other than the late V. I. Arnol’d who grasped the topological angle on these objects, he also being the one who christened them Vassiliev invariants.

The book’s excellent Preface goes on to give an *in embryo* characterization of the objects in the title by noting that

Vassiliev’s definition of finite type invariants is based on the observation that knots form a topological space and knot invariants can be thought of as the locally constant functions on this space. Indeed, the space of knots is an open subspace of the space \(M\) of all smooth maps from \(S^1\) to \mathbb{R}^3\); its complement is the so-called discriminant \(\Sigma\) which consists of all maps that fail to be embeddings. Two knots are isotopic if and only if they can be connected in \(M\) by a path that does not cross \(\Sigma\).

This is a beautiful amalgam of (what?) analysis situs, general topology, analysis proper — some of each, I guess. The authors go on to say:

Using simplicial resolutions, Vassiliev constructs a spectral sequence for the homology of \(\Sigma\). After applying Alexander duality, this spectral sequence produces cohomology classes for the space of knots \(M-\Sigma\); in dimension zero these are precisely the Vassiliev knot invariants.

This certainly indicates a substantial requirement placed on the reader as far as homological algebra and algebraic topology are concerned, and that is no doubt as it should be. But there is a lot of good news for boots-on-the-ground low-dimensional topologists: Chmutov, Duzhin, and Mostovoy go on to note, first, that in 1993 Birman and X.-S. Lin produced the result mentioned above, connecting Vassiliev invariants and quantum group invariants including the Jones polynomial, and, second, that in the same year Maxim Kontsevich showed that the theory of Vassiliev invariants taking values in \(\mathbb{R}\) “can … be reduced entirely to the combinatorics of chord diagrams” (using the yoga of Kontsevich integrals). Subsequently they mention the work of Dror Bar-Natan concerning “the relationship between finite-type invariants and the topological quantum field theory developed by … E. Witten.” The book under review does not cover the last mentioned theme, and the authors refer instead to a number of (seminal) papers. They point the reader to an online bibliography maintained by Bar-Natan.

The last comments of the Preface are these: “The book is intended to be a textbook, so we have included many exercises. Some .. are embedded in the text; the others appear in a separate section at the end of each chapter. Open problems are marked with an asterisk [since this is after all an evolving subject].”

As being a textbook — and an excellent one — the authors take us from a dense but accessible introduction to knots as such to quantum invariants, all in the first two chapters, and then go on to Vassiliev’s finite type invariants. Then we get to chord diagrams, Lie algebra connections, Kontsevich’s integral, work by Drinfel’d, more stuff on the Kontsevich integral, material on braids, and more. The book closes with a chapter on “[t]he space of all knots” (see above). It’s very, very attractrive material.

By the way, in the earlier Preface, Chmutov, Duzhin, and Mostovoy observe that “the most important question put forward in 1990: *Is it true that Vassiiev invariants distinguish knots* [,] is still open.” The book under review carries a 2012 copyright: what about today? Well, according to Wikipedia, we’re still out of luck. So: onward!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.