Computation of the Bennequin invariant of a Legendre curve from the geometry of its front

Abstract

i. Let M s be a contact manifold and ~ a contractible Legendre curve in it. The Bennequin invariant £(7) is defined as the linking coefficient of 7 with the curve obtained by a small shift of ~ along a transversel to it in contact planes [i]. We shall compute the Bennequin invariant in the following contact manifolds: ~I1~S I × R 2 , the manifold of unoriented contact elements; M~S ~ >~ R2, its n-sheeted covering; M ~ ~ , its universal covering. If x, y are coordinates in ~ and ~ is the angular coordinate, then the contact form is I = sin ~dxcos ~dy.