No abstract

Robertson, Seymour, and Thomas characterized linkless embeddings of graphs by flat embeddings, and determined the obstruction set for linkless embeddings. In this paper, we extend flat embeddings to "primitive embeddings" as linkless embeddings to knotless embeddings. Although the obstruction set for knotless embeddings has not been determined, fundamental theorems and conjectures are obtained.

Applying Jaco's Handle Addition Lemma, we give a condition for a 3-manifold to have an incompressible boundary. As an application, we show that the boundary of the exterior of a minimally knotted planar graph is incompressible.

A universal bound on the number of mutually disjoint non-parallel genus one Seifert surfaces for hyperbolic knots in non-Haken manifolds is given.

Abstract. We show that if there exists an essential accidental surface in the knot exterior, then a closed accidental surface also exists. As its corollary, we know boundary slopes of accidental essential surfaces are integral or meridional. It is shown that an accidental incompressible Seifert surface in knot exteriors in S 3 is totally knotted. Examples of satellite knots with arbitrarily high genus Seifert surfaces with accidental peripherals are given, and a Haken 3-manifold which contains a hyperbolic knot with an accidental incompressible Seifert surface of genus one is also given.