An algorithm of finding planar surfaces in three-manifolds

E. A. Sbrodova

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 2 (2005), pp. 192-193.

Résumé

This paper is devoted to the question: does there exist an algorithm to decide whether or not a given $3$-manifold contains a proper essential planar surface? By a planar surface we mean a punctured disc. There is an algorithm, due to W. Jaco, to decide whether a $3$-manifold admits a proper essential disc, i.e., whether it is boundary reducible. A close result, an algorithm allow us to say whether a manifold contains a proper essential disc with a given boundary, was obtained by W. Haken in 60-th. In 1998 W. Jaco, H. Rubinstein and E. Sedgwick described an algorithm to decide whether or not a given linkmanifold contains a proper essential planar surface (a link-manifold is a compact orientable $3$-manifold whose boundary consists of tori) [1]. We generalize this result to manifolds with arbitrary boundaries. A slope on the boundary of a $3$-manifold $M$ is the isotopy class of a finite set of disjoint simple closed curves $\{\alpha_1,\dots,\alpha_n\}$ in $\partial M$ which are nontrivial and pairwise nonparallel. We say that the boundary of a proper surface $F$ has a slope $\alpha=\{\alpha_1,\dots,\alpha_n\}$ if the boundary components of $F$ are each parallel to one of the curves $\alpha_1,\dots,\alpha_n$.

Bibliographie

[1] Jaco W. and Sedgwick E., Decision problems in the space of Dehn fillings, arXiv: math.GT/9811031 | MR

[2] Matveev S., Algoritmic Topology and Classification of 3-Manifolds, Springer-Verlag, Berlin, Heidelberg, 2003 | MR