An explicit formula for a Bennequin-type invariant of apparent contours (Q1005169)

Given a smooth surface, and its generic smooth map to the plane, the apparent contour is the branching locus of the map, that is, the subset of the plane consisting of points that have preimages in the surface where the map is singular. The apparent contour typically consists of smooth curves with transversal crossings and cusps. Ohmoto and Aicardi computed the space of local, first order Vassiliev-invariants of apparent contours. The space turns out to be three dimensional. Two generators have simple interpretations in terms of the number of crossings and cusps. The third invariant is defined through the Legendrian lifting of the contour, and its Bennequin invariant. In the paper under review the authors describe a recipe to read the value of this third Aicardi-Ohmoto invariant from the contour without constructing the Legendrian lift. The description uses only the orientation, the nodes and cusps, as well as the extremal points of the contour. Illuminating examples and computer codes are also included.