Mathematics

A front is a plane curve whose singularities are only transverse self-intersections and semi-cubic cusps and whose tangent is nowhere vertical. Closed fronts are plane projections of Legendrian knots, i.e. knots in R^3 lying in the standard contact distribution du-pdq=0. Two Legendrian knots are said to be Legendrian equivalent, if one can be deformed to another by a continuous path in the space of Legendrian knots. Apart from the usual topological type of the knot in R^3, Legendrian knots have two additional integer invariants, the Maslov number and the Bennequin number. Initially, there was the conjecture that the concidence of these two numbers as well as the usual topological equivalence is sufficient for the Legendrian equivalence. This conjecture was disproved by Yu.Chekanov who invented a rather complicated additional invariant of Legendrian knots (the cohomology ring of a specially constructed differential graded algebra). After only mentioning this fact, we will pass to a more detailed description of a much more elementary construction due to P.Pushkar (\"Pushkar's fences\").