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Link invariants of finite type and perturbation theory. (English) Zbl 0792.57002
Summary: 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 \(V_ \infty\) containing elements \(g_ i\) satisfying the usual braid group relations and elements \(a_ i\) satisfying \(g_ i - g_ i^{-1} = \varepsilon a_ i\), where \(\varepsilon\) is a formal variable that may be regarded as measuring the failure of \(g^ 2_ i\) 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 \(V_ \infty\). 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.

57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
81T13 Yang-Mills and other gauge theories in quantum field theory
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