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

Tsuyoshi Kobayashi and Osamu Saeki

Vol. 195 (2000), No. 1, 101–156

DOI: 10.2140/pjm.2000.195.101

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 non-trivial two bridge knot is obtained from a 2-bridge position in a standard manner.

Milestones

Received: 1 April 1998

Revised: 24 June 1999

Published: 1 September 2000