Stable maps and hyperbolic links

Abstract

A stable map of a closed orientable $3$-manifold into the real plane is called a stable map of a link in the manifold if the link is contained in the set of definite fold points. We give a complete characterization of the hyperbolic links in the $3$-sphere that admit stable maps into the real plane with exactly one (connected component of a) fiber having two singular points.

22 pages, 35 figures; final version, to appear in Comm. Anal. Geom