Invariants of Links and 3-Manifolds from Graph Configurations

This self-contained book explains how to count graph configurations to obtain topological invariants for 3-manifolds and links in these 3-manifolds, and it investigates the properties of the obtained invariants. The simplest of these invariants is the linking number of two disjoint knots in the ambient space described in the beginning of the book as the degree of a Gauss map.

Mysterious knot invariants called “quantum invariants” were introduced in the mid-1980s, starting with the Jones polynomial. Witten explained how to obtain many of them from the perturbative expansion of the Chern–Simons theory. His physicist viewpoint led Kontsevich to a configuration-counting definition of topological invariants for the closed 3-manifolds where knots bound oriented compact surfaces. The book's first part shows in what sense an invariant previously defined by Casson for these manifolds counts embeddings of the theta graph. The second and third parts describe a configuration-counting invariant $\mathcal{Z}$ generalizing the above invariants. The fourth part shows the universality of $\mathcal{Z}$ with respect to some theories of finite-type invariants. The most sophisticated presented generalization of $\mathcal{Z}$ applies to small pieces of links in 3-manifolds called tangles. Its functorial properties and its behavior under cabling are used to describe the properties of $\mathcal{Z}$.

The book is written for graduate students and more advanced researchers interested in low-dimensional topology and knot theory.