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Conway polynomial and Magnus expansion


Author: S. V. Duzhin
Translated by: the author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 541-550
MSC (2010): Primary 20F36
DOI: https://doi.org/10.1090/S1061-0022-2012-01207-0
Published electronically: March 2, 2012
MathSciNet review: 2896167
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Abstract | References | Similar Articles | Additional Information

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.


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  • 1. D. 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; Online at arXiv:q-alg/9607001. MR 1372767 (97b:57004)
  • 2. J. S. Birman, Braids, links and mapping class groups, Ann. of Math. Stud., No. 82, Princeton Univ. Press, Princeton, 1974. MR 0380762 (52:1659)
  • 3. 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.
  • 4. V. Drinfel'd, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $ \mathrm {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$, Algebra i Analiz 2 (1990), no. 4, 149-181; English transl., Leningrad Math. J. 2 (1991), no. 4, 829-860. MR 1080203 (92f:16047)
  • 5. S. Duzhin, Program and data files related to the Drinfeld associator, Online at http:// www.pdmi.ras.ru/~arnsem/dataprog/associator/.
  • 6. P. Etingof and O. Schiffmann, Lectures on quantum groups, Internat. Press, Boston, MA, 1998. MR 1698405 (2000e:17016)
  • 7. R. Graham, D. Knuth, and O. Patashnik, Concrete mathematics. A foundation for computer science, Addison-Wesley, Reading, MA, 1994. MR 1397498 (97d:68003)
  • 8. M. E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), 275-290. MR 1141796 (92i:11089)
  • 9. T. T. Q. Le and J. Murakami, The Kontsevich's integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996), 39-65. MR 1399467 (97d:57009)
  • 10. J. Mostovoy and T. Stanford, On a map from pure braids to knots, J. Knot Theory Ramifications 12 (2003), 417-425; http://www.matcuer.unam.mx/~jacob/works.html. MR 1983095 (2004j:57010)
  • 11. J. Mostovoy and S. Willerton, Free groups and finite-type invariants of pure braids, Math. Proc. Cambridge Philos. Soc. 132 (2002), 117-130; http://www.matcuer.unam.mx/~jacob/works.html. MR 1866328 (2002i:20054)
  • 12. K. Murasugi, Knot theory and its applications, Birkhäuser, Boston, MA, 1996. MR 1391727 (97g:57011)
  • 13. S. Papadima, The universal finite-type invariant for braids, with integer coefficients, Topology Appl. 118 (2002), 169-185. MR 1877723 (2003a:20060)
  • 14. M. Petitot, Tables of relations between MZV up to weight 16, Online at http://www2.lifl.fr /~petitot/.
  • 15. V. V. Prasolov and A. B. Sosinskiĭ, Knots, links, braids and 3-manifolds, MTsNMO, Moscow, 1997; English transl., Transl. Math. Monogr., vol. 154, Amer. Math. Soc., Providence, RI, 1997. MR 1414898 (98i:57018)

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Additional 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

DOI: https://doi.org/10.1090/S1061-0022-2012-01207-0
Keywords: Conway polynomial, Magnus expansion, Vassiliev invariants
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.
Article copyright: © Copyright 2012 American Mathematical Society

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