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The space of incompressible surfaces in a 2-bridge link complement. (English) Zbl 0672.57006

The authors extend the classification of incompressible surfaces in 2- bridge-knot complements [A. Hatcher and W. Thurston, Invent. Math. 79, 225-246 (1985; Zbl 0602.57002)] to the case of 2-bridge links. The classification is more involved in this case. The authors construct a polyhedron \({\mathcal P}{\mathcal L}(S^ 3-L_{p/q})\) whose rational points correspond bijectively, in a natural way, to the projective isotopy classes of incompressible surfaces in the exterior of a 2-bridge link \(L_{p/q}\subset S^ 3\). To construct \({\mathcal P}{\mathcal L}(S^ 3- L_{p/q})\) they first find a fairly natural finite collection of branched surfaces \(B_ i\subset S^ 3-L_{p/q}\) which carry all the incompressible surfaces in \(S^ 3-L_{p/q}\). They expect that \({\mathcal P}{\mathcal L}(S^ 3-L_{p/q})\) is the projective lamination space of \(S^ 3-L_{p/q}\). It extends Thurston’s theory of projective lamination spaces from surfaces to 3-manifolds [W. Thurston, Bull. Am. Math. Soc., New Ser. 19, No.2, 417-431 (1988)].
Reviewer: J.Przytycki

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)

Citations:

Zbl 0602.57002
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