, Volume 26, Issue 1, pp 43-51

Link invariants of finite type and perturbation theory

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Abstract

The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V x containing elements g i satisfying the usual braid group relations and elements a i satisfying g ig infi sup-1 a i, where ε is a formal variable that may be regarded as measuring the failure of g infi sup2 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 x. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphisminvariant perturbation theory for quantum gravity in the loop representation.