Knot invariants derived from the equivariant linking pairing

Details Knot invariants derived from the equivariant linking pairing Knot invariants derived from the equivariant linking pairing

2010-01-25

arxiv.org/abs/1001.4474v3

Mathematics

Geometric Topology

26 pages, uses pstricks, revised version, construction of invariants adjusted to make them actually live in the target space o

Scientific paper

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 H_1(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 equivalent 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.

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