Link Invariants of Finite Type and Perturbation Theory

TL;DR

This paper explores the algebraic structure underlying finite type link invariants and their relation to perturbation theory, proposing a new algebraic framework and potential applications to quantum gravity.

Contribution

Introduction of the algebra V_infinity that encodes braid relations and crossings, linking finite type invariants to Markov traces and perturbation theory.

Findings

Finite type link invariants correspond to Markov traces on V_infinity.

The algebra V_infinity encodes braid relations and crossing information.

Potential application to diffeomorphism-invariant quantum gravity perturbation theory.

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_infinity containing elements g_i satisfying the usual braid group relations and elements a_i satisfying g_i - g_i^{-1} = epsilon a_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 crossings. 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.