Combinatorial quantum method in 3-dimensional topology (Q1295981)

Finite type and quantum invariants of knots and 3-manifolds have become one of the topics of prime interest in low-dimensional topology. Yet it is difficult to get acquainted with the subject. There are different approaches, widely spread in a number of research papers, but no clear introduction, not even a common vocabulary. The book under review, based on a series of lectures held by the author in a workshop with the same title at Waseda University in 1996, might help to fill this gap. It is not an introductory textbook, though, which gives long explanations and proofs in all detail. The reader should have a rough idea what finite type and quantum invariants are about. On the other hand, by often only sketching the idea and some central details of proofs, the author manages to provide the reader with an overview, a good feeling of ``what is happening'', which is quite valuable. For the reader who wants to go more into detail, bibliographical reference is always given. As the title suggests, the author tries a ``unifying'' approach to the topic. The first half of the book treats invariants of knots, the second half invariants of 3-manifolds, both expositions following the same pattern: first, a ``universal'' invariant is introduced, then its relation to quantum invariants and finite type invariants is investigated. In the case of knots, the universal invariant is the well-known modified Kontsevich-invariant, which is introduced in a combinatorial fashion, using non-associative or quasi-tangles and avoiding Kontsevich's integral expression. In the two chapters following this introduction, the author shows how quantum invariants of finite type or Vassiliev invariants recover from this universal invariant, another chapter treats direct relationships between quantum and Vassiliev invariants. In the case of 3-manifolds, not quite as much is known. First, again, a universal invariant is introduced. This is the universal perturbative invariant of 3-manifolds, or LMO-invariant, defined by the author, \textit{T. T. Q. Le} and \textit{J. Murakami} [Topology 37, No. 3, 539-574 (1998; Zbl 0897.57017)]. Large parts of the remaining half of the book are based on this paper, already the technical preliminaries for the invariant's definition make up a fourth part of the whole book. (Not surprisingly, this invariant is the Kontsevich integral, made invariant under Kirby-moves.) The next chapter is dedicated to finite type invariants of integral homology 3-spheres [the author, J. Knot Theory Ramifications 5, No. 1, 101-115 (1996; Zbl 0942.57009)]. It is shown that those, analogously to Vassiliev invariants of knots and links, recover from the universal invariant. The last chapter of the book treats the relationship between quantum invariants of 3-manifolds and the LMO-invariant. Here, a complete result parallel to the case of knots and links does not exist yet. The author quotes some results concerning quantum SO(3)-invariants of rational homology 3-spheres [\textit{H. Murakami}, Math. Proc. Camb. Philos. Soc. 117, No. 2, 237-249 (1995; Zbl 0854.57016); the author, Invent. Math. 123, No. 2, 241-257 (1996; Zbl 0855.57016)]. Parallel results for the quantum \(\text{PSU} (N)\)-invariant [\textit{T. Kohno} and \textit{T. Takata}, J. Knot Theory Ramifications 2, No. 2, 149-169 (1993; Zbl 0795.57004)] are conjectured. Most of the chapter is dedicated to showing that for \(N=2\), this invariant can be recovered from the universal perturbative invariant. What gives the book its special flavour is first its analogous treatment of knots and 3-manifolds, and second its unifying approach to the different types of invariants.