Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves |
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| Authors: | S Chmutov V Goryunov and H Murakami |
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| Institution: | (1) Program Systems Institute, Pereslavl-Zalessky, 152140 Russia (E-mail: chmutov@math.botik.yaroslavl.su) , RU;(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) Department of Mathematics, Osaka City University, 3-138, Sugimoto 3-chome, Sumiyoshi-ku, Osaka 558, Japan , JP |
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| Abstract: | We show that every unframed knot type in 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.
Received: 17 April 1996 / Revised: 12 May 1999 / Published online: 28 June 2000 |
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| Keywords: | Mathematics Subject Classification (1991): 57M25 53C15 |
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