Past lectures

Abstract

Using a pinched hodographic construction one associates to a generic relative immersion of a 1-manifold in the 2-disk a classical link in the 3-sphere. Especially, to a generic relative immersion of the interval in the disk is associated a classical knot. The knots and links obtained in this way have beautifull properties and generalize the local links of complexe plane curve singularities. For instance, those links are transversal to the natural contact structure of the 3-sphere in such a way that the Bennequin inequality becomes an equality. The unknotting number of the knot of an immersed interval is just the number of double points of the immersion. As for links of singularities, the link of a connected immersion is a fibred link and its monodromy has extremal properties. For slalom immersion around rooted trees in the disk the associated knot is either the knot of an kleinian singularity or a hyperbolic knot. We will especially study the hyperbolic knots of rooted trees with only nodes of valency 2 beside 1 node of valency 3.