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. 
@article{SEMR_2006_3_a29,
     author = {E. A. Sbrodova},
     title = {Planar surfaces in three-manifolds},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {451--463},
     publisher = {mathdoc},
     volume = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2006_3_a29/}
}
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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/