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Mathematische Annalen

, Volume 317, Issue 3, pp 389–413 | Cite as

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

  • S. Chmutov
  • V. Goryunov
  • H. Murakami
Original article

Abstract.

We show that every unframed knot type in \(ST^*{\bf \mathrm{R}}^2\) has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by V.I.Arnold in [3]. The Legendrian lifting lowers the framed version of the HOMFLY polynomial [20] to generic plane curves. We prove that the induced polynomial invariant can be completely defined in terms of plane curves only. Moreover it is a genuine, not Laurent, polynomial in the framing variable. This provides an estimate on the Bennequin-Tabachnikov number of a Legendrian knot.

Mathematics Subject Classification (1991): 57M25,53C15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • S. Chmutov
    • 1
  • V. Goryunov
    • 2
  • H. Murakami
    • 3
  1. 1.Program Systems Institute, Pereslavl-Zalessky, 152140 Russia (E-mail: chmutov@math.botik.yaroslavl.su) RU
  2. 2.Department of Mathematical Sciences, Division of Pure Mathematics, The University of Liverpool, Liverpool L69 3BX, UK (E-mail: goryunov@liv.ac.uk) GB
  3. 3.Department of Mathematics, Osaka City University, 3-138, Sugimoto 3-chome, Sumiyoshi-ku, Osaka 558, Japan JP

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