A topological interpretation for the polar quotients of an algebraic plane curve singularity

Rather than a survey, this article is a guided tour into the maze of polars, organized for the benefit of topologically minded readers. It is written for a person at home with 3-dimensional topology. On the other hand, very little is assumed from the field of singularities. To each germ of curve V at the origin of ℂ2 there is attached a pencil of curves : the pencil of polars. The topological type of its "general fiber" is a local analytic invariant of V. It is "the" polar of V. The set of contact quotients (in H. Hironaka's sense) of V with the branches of the polar is known to be an invariant of the local topological type of V. We show here how to interpret this set of polar quotients in terms of the minimal Waldhausen decomposition of the exterior of the (algebraic) link in S3 associated to V.