The Rubinstein–Scharlemann graphic of a 3-manifold as the discriminant set of a stable map

Abstract

We show that Rubinstein-Scharlemann graphics for 3-manifolds can be regarded as the images of the singular sets (: discriminant set) of stable maps from the 3-manifolds into the plane. As applications of our understanding of the graphic, we give a method for describing Heegaard surfaces in 3-manifolds by using arcs in the plane, and give an orbifold version of Rubinstein-Scharlemann's setting. Then by using this setting, we show that every genus one 1-bridge position of a nontrivial two bridge knot is obtained from a 2-bridge position in a standard manner.

Parameterizations of 1-Bridge Torus Knots

“…Using [17], they characterized the intersection when M admits a double cover branched along K. In the next section we will give a necessary and sufficient condition for the existence of this double cover in terms of parameters in a Schubert's normal form.…”

Automorphisms of the three-torus preserving a genus-three Heegaard splitting

“…We can think of π as a circle valued Morse function on T 3 . As was shown in [Kobayashi and Saeki 2000], after an arbitrarily small isotopy the product of two Morse functions on a 3-manifold M is a stable function from M to ‫ޒ‬ 2 . The same argument in this situation implies that after an arbitrarily small isotopy, the product of f and π is a stable function f × π : T 3 → [0, 1] × S 1 .…”

“…Define π t to be the restriction of π to f t . As was shown in [Kobayashi and Saeki 2000], the discriminant set of F is a one dimensional submanifold in T 3 , whose image F() is a finite graph in [0, 1] × S 1 , called the graphic of f . We will say that F is generic if F is stable and for any s ∈ S 1 , there is at most one vertex of F() in the line [0, 1] × {s}.…”

Surface bundles versus Heegaard splittings

“…This technique is also similar that of [6]. We also refer the reader to [10] as an informative paper on this topic.…”