An introduction to finite type invariants of knots and 3-manifolds

Christine Lescop (1)

1 IF - Institut Fourier

These notes are the notes of five lectures presented in the ICPAM-ICTP research school of Meknès in May 2012, as an introduction to finite type invariants of links and 3-manifolds. The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. For 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 that is the Casson invariant of integral homology spheres. It is defined as the algebraic intersection of three propagators in a 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 a construction based on configuration space integrals of universal finite type invariants for links in rational homology spheres and we state open problems. In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.

Mots clés: 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

Domaines: Topologie géométrique [math.GT]

Christine Lescop. An introduction to finite type invariants of knots and 3-manifolds. 2012. ⟨hal-00915950v1⟩