{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T19:40:16Z","timestamp":1735760416692,"version":"3.32.0"},"reference-count":32,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Knot Theory Ramifications"],"published-print":{"date-parts":[[2005,6]]},"abstract":"<jats:p>We analyse the possibility of defining \u2102-valued Knot invariants associated with infinite-dimensional unitary representations of SL(2,\u211d) and the Lorentz Group taking as starting point the Kontsevich integral and the notion of infinitesimal character. This yields a family of knot invariants whose target space is the set of formal power series in \u2102, which contained in the Melvin\u2013Morton expansion of the coloured Jones polynomial. We verify that for some knots the series have zero radius of convergence and analyse the construction of functions of which this series are asymptotic expansions by means of Borel re-summation. Explicit calculations are done in the case of torus knots which realise an analytic extension of the values of the coloured Jones polynomial to complex spins. We present a partial answer in the general case.<\/jats:p>","DOI":"10.1142\/s0218216505003889","type":"journal-article","created":{"date-parts":[[2005,6,16]],"date-time":"2005-06-16T11:24:12Z","timestamp":1118921052000},"page":"435-466","source":"Crossref","is-referenced-by-count":1,"title":["ON THE ANALYTIC PROPERTIES OF THE z-COLOURED JONES POLYNOMIAL"],"prefix":"10.1142","volume":"14","author":[{"given":"JO\u00c3O FARIA","family":"MARTINS","sequence":"first","affiliation":[{"name":"Departamento de Mathem\u00e1tica, Instituto Superior T\u00e9cnico, Av Rovisco Pais, 1049-001, Lisboa, Portugal"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","first-page":"262","volume":"127","author":"Reshetikhin N. Y.","journal-title":"Commun. Math. Phys."},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(89)90018-2"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099116"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/17\/16\/302"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/19\/19\/313"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/21\/2\/009"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/20\/14\/318"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(95)93237-2"},{"volume-title":"Quantum Groups","year":"1994","author":"Kassel C.","key":"rf11"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(95)00071-2"},{"key":"rf13","first-page":"41","volume":"102","author":"Le T.","journal-title":"Comp. Math."},{"key":"rf14","first-page":"1","volume":"7","author":"Le T.","journal-title":"Geom. Topol."},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1016\/S0393-0440(98)00007-2"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66243-0"},{"key":"rf17","first-page":"1419","volume":"1","author":"Drinfeld V. G.","journal-title":"Leningrad Math. J."},{"key":"rf18","first-page":"829","volume":"2","author":"Drinfeld V. G.","journal-title":"Leningrad Math. J."},{"volume-title":"SL(2; \u211d)","year":"1975","author":"Lang S.","key":"rf19"},{"key":"rf20","unstructured":"I. M.\u00a0Gel'fand, M. I.\u00a0Graev and N. Y.\u00a0Vilenkin, Generalized Functions 5, Integral Geometry and Representation Theory (Academic Press, New York, 1966)\u00a0p. xvii."},{"volume-title":"Lie groups, Lie Algebras, and Their Representations","year":"1974","author":"Varadarajan V. S.","key":"rf21"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099310"},{"key":"rf23","first-page":"103","volume":"125","author":"Garoufalidis S.","journal-title":"Invent. Math."},{"key":"rf24","doi-asserted-by":"crossref","unstructured":"K.\u00a0Habiro, Invariants of Knots and 3-Manifolds, Geometry and Topology Monographs\u00a04 (Kyoto, 2001)\u00a0pp. 55\u201368.","DOI":"10.2140\/gtm.2002.4.55"},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1007\/BF02473358"},{"key":"rf26","doi-asserted-by":"publisher","DOI":"10.1007\/s002200050736"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1007\/BF02096620"},{"volume-title":"Representations of the Rotation and Lorentz Groups and Their Applications","year":"1963","author":"Gel'fand I. M.","key":"rf28"},{"key":"rf29","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022608131142"},{"volume-title":"Borel\u2013Laplace Transform and Asymptotic Theory","year":"1996","author":"Sternin B.","key":"rf30"},{"key":"rf32","doi-asserted-by":"publisher","DOI":"10.1142\/9789812792679_0034"},{"key":"rf33","doi-asserted-by":"publisher","DOI":"10.1023\/A:1010773818312"},{"key":"rf34","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100073965"},{"key":"rf35","doi-asserted-by":"publisher","DOI":"10.1007\/BF02101842"}],"container-title":["Journal of Knot Theory and Its Ramifications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218216505003889","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T19:07:56Z","timestamp":1735758476000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218216505003889"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,6]]},"references-count":32,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2011,11,21]]},"published-print":{"date-parts":[[2005,6]]}},"alternative-id":["10.1142\/S0218216505003889"],"URL":"https:\/\/doi.org\/10.1142\/s0218216505003889","relation":{},"ISSN":["0218-2165","1793-6527"],"issn-type":[{"type":"print","value":"0218-2165"},{"type":"electronic","value":"1793-6527"}],"subject":[],"published":{"date-parts":[[2005,6]]}}}