Singularities cancellation on wave fronts

We show that any Legendre knot in the contact manifold of cooriented contact elements of a surface M is, up to stabilization, Legendre-isotopic to a Legendre knot whose projection on M (wave front) is an immersion, provided that it is Legendre-homotopic to such a knot. As a consequence, we obtain that each ambient isotopy class of knots contains Legendre representatives with immersed wave fronts. We also show that similar results do not hold in the context of the manifold of noncooriented contact elements.