Invariants of links and 3-manifolds that count graph configurations

Lescop, Christine

Winter Braids X (Pisa, 2020), Winter Braids Lecture Notes (2020), Exposé no. 1, 35 p.

We present ways of counting configurations of uni-trivalent Feynman graphs in 3-manifolds in order to produce invariants of these 3-manifolds and of their links, following Gauss, Witten, Bar-Natan, Kontsevich and others. We first review the construction of the simplest invariants that can be obtained in our setting. These invariants are the linking number and the Casson invariant of integer homology 3-spheres. Next we see how the involved ingredients, which may be explicitly described using gradient flows of Morse functions, allow us to define a functor on the category of framed tangles in rational homology cylinders. Finally, we describe some properties of our functor, which generalizes both a universal Vassiliev invariant for links in the ambient space and a universal finite type invariant of rational homology 3-spheres.

Classification : 57K16, 57K31, 57K30, 55R80, 57R20, 81Q30
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

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