The two-eigenvalue problem and density of Jones representation of braid groups

In 1983 V. Jones discovered a new family of representations ρ of the braid groups. They emerged from the study of operator algebras (type Π1 factors) and unlike earlier braid representations had no naive homological interpretation. Almost immediately he found that the trace or “Markov” property of ρ allowed new link invariants to be defined and this ushered in the era of quantum topology. There has been an explosion of link and 3- manifold invariants with beautiful inter-relations, asymptotic formulae, and enchanting connections to mathematical physics: Chern-Simons theory and 2-dimensional statistical mechanics. While many sought to bend Jones’ theory toward classical topological objectives, we have found that the relation between the Jones polynomial and physics allows potentially realistic models of quantum computation to be created [FKW][FLW][FKLW][F]. Unitarity, a hidden locality, and density of the Jones representation are central to computational applications. With this application in mind, we have returned to some of Jones’ earliest questions about these representations and the distributions of his invariants. A few concise answers are stated here in the introduction. Question 9 of Jones in [J2] asked for the closed images of the irreducible components of his representation. We answer Jones’ question, and also identified the closed images for the general SU(N) case completely.