Quantum Invariants of Knots and 3-Manifolds

In hindsight, the discovery in 1984 of the Jones polynomial was the beginning of the new subject of quantum topology. I've never seen a precise definition of “quantum topology” but the notion of “Topological Quantum Field Theory” (TQFT) is certainly of central concern. Roughly speaking, an (n+1)-dimensional TQFT consists of:

1. An assignment of a vector space V(F) to each compact, oriented n-dimensional manifold
2. An assignment of a linear map L(M):V(F) → V(G) to each oriented (n+1)-dimensional manifold M bounded by n-manifolds F and G.

Of course, it is required that certain axioms be satisfied. For example: the 0-dimensional vector space is assigned to the empty set, thought of as an n-dimensional manifold; and the vector space associated to the disjoint union of two n-dimensional manifolds F and G is required to be the tensor product of the vector spaces assigned to F and G.

If the vector spaces are vector spaces over the complex numbers C, then associated to each closed (n+1)-dimensional manifold M is a linear map L(M):C → C. Such a map is determined completely by where it sends 1 ∈ C and so L(M)(1) is a complex number that is an invariant of M.

The history of topology shows that we should get excited whenever we can construct a non-trivial invariant of M. We would hope, therefore, that we could construct a TQFT such that the invariants produced by the TQFT are:

1. Non-trivial: i.e. there exist manifolds with distinct invariants
2. Topologically useful: i.e. knowing the invariant of a manifold can tell us something interesting about its topology.
3. Calculable: Given a description of the manifold (for example, a cell decomposition), it should be possible to compute the invariant of the manifold in a reasonable (eg. finite) amount of time.

The purpose of the book under review is to construct non-trivial, topologically useful, and calculable (2+1)-dimensional TQFTs.

The book is carefully and clearly written and has become a classic in the field. For me, it is also an extremely challenging read. It relies heavily on a fair amount of category theory and much of the book is taken up with algebraic (rather than topological) results.

This book is undoubtedly extremely useful for those doing research in quantum topology. (If you, dear reader, are an object in the category of quantum topologists then you already knew you should read this book.) Those seeking only some exposure to quantum topology, would do well to focus on the introduction and chapters II.2, III.1, III.2, IV.1, VII.3, and chapter XII. A clear exposition of (a less general form of) the material in chapter XII can also be found in a number of papers of Raymond Lickorish. The book is now in its second edition; according to the author, “The second edition does not essentially differ from the first one.”

Scott Taylor is a knot theorist and 3-manifold topologist who enjoys observing quantum topology from afar. He is an assistant professor at Colby College.