Note on the degree of $C^0$-sufficiency of plane curves

Abstract

Let f be a germ of plane curve, we define the 8-degree of sufficiency offto be the smallest integer r such that for any germ g such that j(r)f = j(r)g then there is a set of disjoint annuli in S3 whose boundaries consist of a component of the link of f and a component of the link of g. We establish a formula for the 8-degree of sufficiency in terms of link invariants of plane curves singularities and, as a consequence of this formula, we obtain that the 8-degree of sufficiency is equal to the C° -degree of sufficiency