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.

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