On the cube of the equivariant linking pairing for knots and 3--manifolds of rank one

Christine Lescop (1)

1 IF - Institut Fourier

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 space of ordered pairs of distinct points of M. We show how to define the equivariant cube Q(K) of this Blanchfield pairing with respect to a framed knot K that generates H_1(M)/Torsion. This article is devoted to the study of the invariant Q. We prove many properties for this invariant including two surgery formulae. Via surgery, the invariant Q is equivalent to an invariant of null-homologous knots in rational homology spheres, that coincides with the two-loop part of the Kricker rational lift of the Kontsevich integral, at least for knots with trivial Alexander polynomial in integral homology spheres.

Mots clés: configuration space integrals; finite type invariants of knots and 3-manifolds; homology spheres; two-loop polynomial; rational lift of Kontsevich integral; equivariant Blanchfield linking pairing; Casson-Walker invariant; LMO invariant; clasper calculus; Jacobi diagrams; perturbative expansion of Chern-Simons theory; surgery formula

Domaines: Topologie géométrique [math.GT]

Christine Lescop. On the cube of the equivariant linking pairing for knots and 3--manifolds of rank one. 2010. ⟨hal-00512360⟩