Conway polynomial and Magnus expansion
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S. V. Duzhin
Translated by: the author - St. Petersburg Math. J. 23 (2012), 541-550
- DOI: https://doi.org/10.1090/S1061-0022-2012-01207-0
- Published electronically: March 2, 2012
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Abstract:
The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short-circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. In the paper, the resulting mapping from horizontal chord diagrams on 3 strands to univariate polynomials is described explicitly and evaluated on the Drinfeld associator, which leads to a beautiful generating function whose coefficients are, conjecturally, alternating sums of multiple zeta values.References
- Dror Bar-Natan, Vassiliev and quantum invariants of braids, The interface of knots and physics (San Francisco, CA, 1995) Proc. Sympos. Appl. Math., vol. 51, Amer. Math. Soc., Providence, RI, 1996, pp. 129–144. MR 1372767, DOI 10.1090/psapm/051/1372767
- Joan S. Birman, Mapping class groups of surfaces: a survey, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974, pp. 57–71. MR 0380762
- S. Chmutov, S. Duzhin, and J. Mostovoy, Introduction to Vassiliev knot invariants (to appear in Cambridge Univ. Press); draft available online at http://arxiv.org/abs/1103.5628.
- V. G. Drinfel′d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $\textrm {Gal}(\overline \textbf {Q}/\textbf {Q})$, Algebra i Analiz 2 (1990), no. 4, 149–181 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 4, 829–860. MR 1080203
- S. Duzhin, Program and data files related to the Drinfeld associator, Online at http:// www.pdmi.ras.ru/~arnsem/dataprog/associator/.
- Pavel Etingof and Olivier Schiffmann, Lectures on quantum groups, Lectures in Mathematical Physics, International Press, Boston, MA, 1998. MR 1698405
- Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
- Michael E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275–290. MR 1141796
- Thang Tu Quoc Le and Jun Murakami, Kontsevich’s integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996), 39–65. MR 1399467, DOI 10.1017/S0027763000005638
- Jacob Mostovoy and Theodore Stanford, On a map from pure braids to knots, J. Knot Theory Ramifications 12 (2003), no. 3, 417–425. MR 1983095, DOI 10.1142/S021821650300255X
- Jacob Mostovoy and Simon Willerton, Free groups and finite-type invariants of pure braids, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 1, 117–130. MR 1866328, DOI 10.1017/S0305004102005868
- Kunio Murasugi, Knot theory and its applications, Birkhäuser Boston, Inc., Boston, MA, 1996. Translated from the 1993 Japanese original by Bohdan Kurpita. MR 1391727
- Ştefan Papadima, The universal finite-type invariant for braids, with integer coefficients, Topology Appl. 118 (2002), no. 1-2, 169–185. Arrangements in Boston: a Conference on Hyperplane Arrangements (1999). MR 1877723, DOI 10.1016/S0166-8641(01)00049-9
- M. Petitot, Tables of relations between MZV up to weight 16, Online at http://www2.lifl.fr /~petitot/.
- V. V. Prasolov and A. B. Sossinsky, Knots, links, braids and 3-manifolds, Translations of Mathematical Monographs, vol. 154, American Mathematical Society, Providence, RI, 1997. An introduction to the new invariants in low-dimensional topology; Translated from the Russian manuscript by Sossinsky [Sosinskiĭ]. MR 1414898, DOI 10.1090/mmono/154
Bibliographic Information
- S. V. Duzhin
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
- Email: duzhin@pdmi.ras.ru
- Received by editor(s): October 3, 2010
- Published electronically: March 2, 2012
- Additional Notes: Supported by grants RFBR 08-01-00379-a, NSh 709.2008.1, and JSPS S-09018.
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 541-550
- MSC (2010): Primary 20F36
- DOI: https://doi.org/10.1090/S1061-0022-2012-01207-0
- MathSciNet review: 2896167