Laboratoire de Topologie et Géométrie Emile Picard, Université Paul Sabatier, M.I.G. 118, Route de Narbonne, 31062 Toulouse-Cedex, France
Abstract:
In this paper we define invariants under smooth isotopy for certain two-dimensional knots using some refined Cerf theory. One of the invariants is the knot type of some classical knot generalizing the string number of closed braids. The other invariant is a generalization of the unique invariant of degree 1 for classical knots in 3-manifolds. Possibly, these invariants can be used to distinguish smooth embeddings of tori in some 4-manifolds but which are equivalent as topological embeddings.