LINK INVARIANTS OF FINITE-TYPE AND PERTURBATION-THEORY

The Vassiliev-Gusarov fink invariants of finite type are known to be closely related to perturbation theory for Chem-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V(infinity) containing elements g(i) satisfying the usual braid group relations and elements a(i) satisfying g(i) - g(i)-1 = epsilona(i), where epsilon is a formal variable that may be regarded as measuring the failure of g(i)2 to equal 1. Topologically, the elements a(i) signify intersections. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on V(infinity). We sketch a possible application of link invariants of finite type to a manifestly diffeomorphism-invariant perturbation theory for quantum gravity in the loop representation.