In 1983 V. Jones discovered a new family of representations \u03c1 of the
\nbraid groups. They emerged from the study of operator algebras (type \u03a01
\nfactors) and unlike earlier braid representations had no naive homological
\ninterpretation. Almost immediately he found that the trace or \u201cMarkov\u201d
\nproperty of \u03c1 allowed new link invariants to be defined and this ushered in
\nthe era of quantum topology. There has been an explosion of link and 3-
\nmanifold invariants with beautiful inter-relations, asymptotic formulae, and
\nenchanting connections to mathematical physics: Chern-Simons theory and
\n2-dimensional statistical mechanics. While many sought to bend Jones\u2019 theory
\ntoward classical topological objectives, we have found that the relation
\nbetween the Jones polynomial and physics allows potentially realistic models
\nof quantum computation to be created [FKW][FLW][FKLW][F]. Unitarity, a
\nhidden locality, and density of the Jones representation are central to computational
\napplications. With this application in mind, we have returned
\nto some of Jones\u2019 earliest questions about these representations and the distributions
\nof his invariants. A few concise answers are stated here in the
\nintroduction. Question 9 of Jones in [J2] asked for the closed images of the
\nirreducible components of his representation. We answer Jones\u2019 question,
\nand also identified the closed images for the general SU(N) case completely.<\/p>\n","protected":false},"excerpt":{"rendered":"
In 1983 V. Jones discovered a new family of representations \u03c1 of the braid groups. They emerged from the study of operator algebras (type \u03a01 factors) and unlike earlier braid representations had no naive homological interpretation. Almost immediately he found that the trace or \u201cMarkov\u201d property of \u03c1 allowed new link invariants to be defined […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"MSR-TR-2001-42","msr_organization":"","msr_pages_string":"33","msr_page_range_start":"33","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"Michael J. 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