Abstract

We show that the Bennequin number of a transverse knot in the standard contact 3–space or solid torus is bounded by the negative of the lowest degree of the framing variable in its HOMFLY polynomial. For R3, this fact was established earlier by Fuchs and Tabachnikov [7] by comparison of the results of [1] and [5, 11]. We develop a different, direct approach based on the lowering of the polynomial to transversally framed regular planar curves and the results of [4]. We show, by providing explicit examples, that for knots in R3 with at most 8 double points in their diagrams the estimate is exact.

It is very well known (see, for example, [8]) that any topological knot type in a contact 3–manifold has a Legendrian representative. If the contact structure is coorientable a Legendrian knot gets a natural framing. The question of Legendrian represent ability of an arbitrary framed knot type has in general a negative answer: according to a classical result by Bennequin [1], the self-linking numbers (called in this case also Bennequin numbers) of all canonically framed Legendrian representatives of a fixed unframed knot type in R3 are bounded from one side.

Paper [1] gives an estimate for the Bennequin number in the standard contact 3-space in terms of the genus of a knot. This has a disadvantage of being insensitive to passing from a knot to its mirror image.