Geodesic
Planar surfaces in three-manifolds
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 3 (2006), pp. 451-463.

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In this paper we consider the problem of algorithmic finding of a proper essential planar surface in a given irreducible orientable compact $3$-manifold. The method uses the Haken theory of normal surfaces in $3$-manifolds with boundary pattern [5]. The solution is based on an estimate, considered in [3], of the average length of boundary curves of an arbitrary proper essential planar surface and on the notion of a slope in the boundary of an arbitrary manifold that generalizes the notion of a slope on a torus. 
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E. A. Sbrodova. Planar surfaces in three-manifolds. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 3 (2006), pp. 451-463. http://geodesic.mathdoc.fr/item/SEMR_2006_3_a29/

[1] W. Haken, “Theorie der Normalflächen”, Acta Math., German, 105 (1961), 245–375 | MR | Zbl

[2] W. Jaco and U. Oertel, “An algorithm to decide if a 3-manifold is a Haken manifold”, Topology, 23:2 (1984), 195–209 | DOI | MR | Zbl

[3] W. Jaco, J. H. Rubinstein and E. Sedgwick, Finding planar surfaces in knot- and link-manifolds, arXiv: math/0608700 | MR

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

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

[6] E. A. Sbrodova, “Algoritm nakhozhdeniya ploskikh poverkhnostei v trekhmernykh mnogoobraziyakh”, Fundamentalnaya i prikladnaya matematika, 11:4 (2005), 197–202 | MR