The Loop Expansion of the Kontsevich Integral, Abelian Invariants of Knots and S-equivalence

Hidden in the expansion of the Kontsevich integral, graded by loops rather than by degree, is a new notion of finite type invariants of knots, closely related to S-equivalence, and with respect to which the Kontsevich integral is the universal finite type invariant, modulo S-equivalence. In addition, the 2-loop part Q of the Kontsevich integral behaves like an equivariant version of Casson’s invariant, and its “first derivative” is given in terms of linking functions associated to the universal abelian cover of the knot complement. As a result, we obtain a linear relation among the Casson-Walker invariant of cyclic branched covers of knots, residues of the Q-function, and the signature of the knot.