Geometry and Topology

An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations

C. Lescop<sup>abc

a</sup> French National Institute for Scientific Research, Saint-Martin-d'Hères, France
^b University Joseph Fourier, Saint-Martin-d'Hères, France
^c Institut Fourier, Saint-Martin-d'Hères, France

Abstract: The finite type invariant concept for knots was introduced in the 90's in order to classify knot invariants, with the work of Vassiliev, Goussarov and Bar-Natan, shortly after the birth of numerous quantum knot invariants. This very useful concept was extended to $3$-manifold invariants by Ohtsuki. These introductory lectures show how to define finite type invariants of links and $3$-manifolds by counting graph configurations in $3$-manifolds, following ideas of Witten and Kontsevich. The linking number is the simplest finite type invariant for $2$-component links. It is defined in many equivalent ways in the first section. As an important example, we present it as the algebraic intersection of a torus and a $4$-chain called a propagator in a configuration space. In the second section, we introduce the simplest finite type $3$-manifold invariant, which is the Casson invariant (or the $\Theta$-invariant) of integer homology $3$-spheres. It is defined as the algebraic intersection of three propagators in the same two-point configuration space. In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams. In Sections 4 and 5, we sketch an original construction based on configuration space integrals of universal finite type invariants for links in rational homology $3$-spheres and we state open problems. Our construction generalizes the known constructions for links in $\mathbb{R}^3$ and for rational homology $3$-spheres, and it makes them more flexible. In Section 6, we present the needed properties of parallelizations of $3$-manifolds and associated Pontrjagin classes, in details.

Keywords: knots, $3$-manifolds, finite type invariants, homology $3$-spheres, linking number, Theta invariant, Casson-Walker invariant, Feynman Jacobi diagrams, perturbative expansion of Chern–Simons theory, configuration space integrals, parallelizations of $3$-manifolds, first Pontrjagin class.

UDC: 515.163
BBK: B151.5

Language: English