Geometriae Dedicata

, Volume 65, Issue 2, pp 219-245

First online:

On the Bennequin Invariant and the Geometry of Wave Fronts

  • Emmanuel FerrandAffiliated withCentre de Mathématiques, Ecole Polytechnique

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The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J + in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.

contact geometry wave fronts Legendrian knots Vassiliev invarinats plane and spherical curves projective duality.