Invariants of knots and 3–manifolds derived from the equivariant linking pairing

Invariants of knots and 3–manifolds derived from the equivariant linking pairing Christine Lescop Abstract. Let M be a closed oriented 3-manifold with first Betti number one. Its equivariant linking pairing may be seen as a two-dimensional cohomology class in an appropriate infinite cyclic covering of the configuration space of ordered pairs of distinct points of M . We show how to define the equivariant cube Q(M,K) of this Blanchfield pairing with respect to a framed knot K that generates H1(M ; Z)/Torsion. We present the invariant Q(M,K) and some of its properties including a surgery formula. Via surgery, the invariant Q is equivalent to an invariant Q of null- homologous knots in rational homology spheres, that is conjecturally equiva- lent to the two-loop part of the Kontsevich integral. We generalize the construction of Q to obtain a topological construction for an invariant that is conjecturally equivalent to the whole Kricker rational lift of the Kontsevich integral for null-homologous knots in rational homology spheres. 1. Introduction 1.1. Background. The study of 3–manifold invariants built from integrals over configuration spaces started after the work of Witten on Chern-Simons theory in 1989 [Wi], with work of Axelrod, Singer [AS1, AS2], Kontsevich [Ko], Bott, Cattaneo [BC1, BC2, C], Taubes [T].

• kricker rational


• casson invariant


• blowing up


• m˜


• s1 ?


• homology classes generate


• rational homology


• manifold