A Bennequin Number Estimate for Transverse Knots

Summary

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.