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A computation of the Kontsevich integral of torus knots. (English) Zbl 1082.57009
In this paper, the author studies the Kontsevich integral of torus knots. The author constructs a series of diagrams made of circles joined together in tree-like fashion and colored by some special rational functions. It is shown that this series codes exactly the unwheeled rational Kontsevich integral of torus knots and that it behaves simply under branched coverings. The Kontsevich integral is very difficult to compute and up to the present only the precise expression of the Kontsevich integral for the unknot was conjectured and proven in two papers [D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston, Isr. J. Math. 119, 217–237 (2000; Zbl 0964.57010) and D. Bar-Natan, T. Q. Le Thang and D. P. Thurston, Geom. Topol. 7, 1–31 (2003; Zbl 1032.57008)]. Here the author gives a complete formula for a family of nontrivial knots extending the authors own work in [J. Marché, Topology Appl. 143, No. 1–3, 15–26 (2004; Zbl 1055.57013)].

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
Full Text: DOI EMIS EuDML arXiv
[1] D Bar-Natan, S Garoufalidis, L Rozansky, D P Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot, Israel J. Math. 119 (2000) 217 · Zbl 0964.57010 · doi:10.1007/BF02810669
[2] D Bar-Natan, T T Q Le, D P Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. 7 (2003) 1 · Zbl 1032.57008 · doi:10.2140/gt.2003.7.1 · emis:journals/UW/gt/GTVol7/paper1.abs.html · eudml:123314 · arxiv:math/0204311
[3] S Garoufalidis, Whitehead doubling persists, Algebr. Geom. Topol. 4 (2004) 935 · Zbl 1082.57006 · doi:10.2140/agt.2004.4.935 · emis:journals/UW/agt/AGTVol4/agt-4-40.abs.html · eudml:125677 · arxiv:math/0003189
[4] S Garoufalidis, A Kricker, Finite type invariants of cyclic branched covers, Topology 43 (2004) 1247 · Zbl 1081.57010 · doi:10.1016/j.top.2001.12.001
[5] S Garoufalidis, A Kricker, A rational noncommutative invariant of boundary links, Geom. Topol. 8 (2004) 115 · Zbl 1075.57004 · doi:10.2140/gt.2004.8.115 · emis:journals/UW/gt/GTVol8/paper4.abs.html · eudml:125303 · arxiv:math/0105028
[6] A Kricker, The lines of the Kontsevich integral and Rozansky’s rationality conjecture · arxiv:math.GT/0005284
[7] C Lescop, Introduction to the Kontsevich integral of framed tangles, technical report, Grenoble Summer School (1999)
[8] J Marché, On Kontsevich integral of torus knots, Topology Appl. 143 (2004) 15 · Zbl 1055.57013 · doi:10.1016/j.topol.2004.01.006
[9] T Ohtsuki, A cabling formula for the 2-loop polynomial of knots, Publ. Res. Inst. Math. Sci. 40 (2004) 949 · Zbl 1097.57016 · doi:10.2977/prims/1145475498
[10] B Patureau-Mirand, Non-injectivity of the Hair map · Zbl 1243.57008 · arxiv:math.GT/0202065
[11] L Rozansky, Higher order terms in the Melvin-Morton expansion of the colored Jones polynomial, Comm. Math. Phys. 183 (1997) 291 · Zbl 0882.57004 · doi:10.1007/BF02506408 · arxiv:q-alg/9601009
[12] L Rozansky, A rationality conjecture about Kontsevich integral of knots and its implications to the structure of the colored Jones polynomial (2003) 47 · Zbl 1022.57007 · doi:10.1016/S0166-8641(02)00053-6
[13] D Thurston, Wheeling: a diagrammatic analogue of the Duflo isomorphism, PhD thesis, University of California, Berkeley (2000) · arxiv:math.QA/0006083
[14] P Vogel, Vassiliev theory, technical report, MaPhySto (2000)
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