On a canonical placement of knots in irreducible 3-manifolds

Abstract

If M is a compact connected orientable irreducible 3-manifold and T is a minimal Jaco–Shalen–Johannson system of tori inside M, we define the pieces of M to be regular neighborhoods of incompressible tori in T∪ ∂M, the components of their complement or regular neighborhoods of Seifert fibres in those components that admit Seifert fibrations. For a given isotopy class K of knots inside M we describe, with some restrictions on M, the set of pieces which contain representatives of K . If the knots of K are not contained in balls, we show that the isotopy class of a representative of K inside a piece P is independent of the chosen representative. To cite this article: P. Popescu-Pampu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 677–682.