Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves
The authors extend parallels between regular plane curves and knots, based on approaches due to Arnold and Vassiliev. It is shown that any unframed knot type in the solid torus \(ST^{*}{\mathbb R}^2\) has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by \textit{V. I.~Arnol'd} [Topological invariants of plane curves and caustics, Univ. Lect. Ser. 5 (1994; Zbl 0858.57001)]. Also, restrictions of the Legendrian framings of such lifts are investigated. It is shown that there is an estimate of these framings which comes from the HOMFLY polynomial of a knot in a solid torus, and this is often stronger than Bennequin's inequality.
scientific article; zbMATH DE number 1502092