The space of incompressible surfaces in a $2$-bridge link complement

Abstract

Projective lamination spaces for 2-bridge link complements are computed explicitly. In this paper we construct a polyhedron P£(S3-Lp/q) whose rational points correspond bijectively, in a natural way, with the isotopy classes of incompressible surfaces in the exterior of a 2-bridge link Lp/q c S3. Here means that we factor out by scalar multiplication taking any number of parallel copies of a surface. (Surfaces are not assumed to be connected.) We expect that P£(S3 Lp/q) will turn out to be the projective lamination space of S3 Lp/q33 as defined and studied for general compact irreducible 3-manifolds in [5 and 12]. An unexpected complication not present in Thurston's theory of lamination spaces for surfaces is the fact that P£(S3 Lp/q) is frequently noncompact, for example for the Whitehead link L3/8 (see Figure 5.4, upper left-hand corner). However, as in the general theory, P£(S3 Lp/q) has a natural compactification P7(53-Lp/q) which is a finite polyhedron. To construct P£(S3-Lp/q)3 we first find a fairly natural finite collection of branched surfaces Bi C S3-Lp/q which carry all the incompressible surfaces in S3Lp/q. To each Bi is associated a convex cell ci whose rational points parametrize the classes of surfaces carried by Bi. DiSerent rational points of Ci can determine isotopic surfaces, however, due to the possibility of pushing parts of surfaces across product regions in the complement of Bi (the analogue of digon regions in the complement of a train track on a surface). This leads to a linear projection Pi: Ci Ci °f Ci onto another convex polyhedral cell ci, such that over the interior of ci, isotopy classes of surfaces coincide with fibers of Pi. However, these isotopy relations may not persist over the boundary of ci. Namely, passing to a face of Ci corresponds to passing to a branched subsurface of Bi, and a nonproduct complementary region of this branched subsurface may be decomposed by Bi into product complementary regions of Bi. In this case, rational points of this face of Ci correspond not to (projective) isotopy classes of incompressible surfaces, but to incompressible surfaces with these limiting phantom isotopy relations. The space P7(53 Lp/q) is foed from the cells Ci by identifying their faces in the most natural way, distinguishing different phantom isotopies between the same sets of surfaces. PZ (S3-Lp/q) consists of the open cells of PS (S3 Lp/q) for which the phantom isotopies are actual isotopies. Received by the editors April 12, 1986. 1980 Mathematics Subject Classification (1985 Revtaion). Primary 57M25.