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.
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Received: 17 April 1996 / Revised: 12 May 1999 / Published online: 28 June 2000
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Chmutov, S., Goryunov, V. & Murakami, H. Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves. Math Ann 317, 389–413 (2000). https://doi.org/10.1007/PL00004407
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DOI: https://doi.org/10.1007/PL00004407