Link Invariants of Finite Type and Perturbation Theory

الأصل · EN

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 Vᵢnfinity containing elements gᵢ satisfying the usual braid group relations and elements aᵢ satisfying gᵢ - gᵢ⁻¹ = epsilon aᵢ, where epsilon is a formal variable that may be regarded as measuring the failure of gᵢ² to equal 1. Topologically, the elements aᵢ signify crossings. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on Vᵢnfinity. 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.