TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 305, Number 2, February 1988 THE SPACE OF INCOMPRESSIBLE SURFACES IN A 2-BRIDGE LINK COMPLEMENT W. FLOYD AND A. HATCHER ABSTRACT. Projective lamination spaces for 2-bridge link complements are computed explicitly. In this paper we construct a polyhedron PC(S3 — Lp/q) whose rational points correspond bijectively, in a natural way, with the projective isotopy classes of in- compressible surfaces in the exterior of a 2-bridge link Lp/q C S3. Here "projective" 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 PC(S3 - Lp/q) will turn out to be the "projective lamination space of S3 - Fp/g" as defined and studied for general compact irreducible 3-manifolds in [5 and 12]. An unexpected complication not present in Thurston's theory of projective lam- ination spaces for surfaces is the fact that PL(S3 — Lv/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, Pi(S3 - Lp/q) has a natural compactification ~P~L(S3 - Lp/q) which is a finite polyhedron. To construct P£,(S3 — Lp/q), we first find a fairly natural finite collection of branched surfaces Bi C S3—Lp/q which carry all the incompressible surfaces in 53 — Lp/q. To each Ft is associated a convex cell c¿ whose rational points parametrize the projective classes of surfaces carried by Bi. Different rational points of c¿ can determine isotopic surfaces, however, due to the possibility of pushing parts of surfaces across product regions in the complement of Ft (the analogue of "digon" regions in the complement of a train track on a surface). This leads to a linear projection Pi'.Ci —* c¿ of c, onto another convex polyhedral cell c¿, such that over the interior of c¿, projective isotopy classes of surfaces coincide with fibers of p¿. However, these isotopy relations may not persist over the boundary of Ci. Namely, passing to a face of c, corresponds to passing to a branched subsurface of F¿, and a nonproduct complementary region of this branched subsurface may be decomposed by Bi into product complementary regions of F¿. 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 ~P~Z(S3 — Lp/q) is formed from the cells c¿ by identifying their faces in the most natural way, distinguishing different "phantom" isotopies between the same sets of surfaces. PC(S3 — Lp/q) consists of the open cells of P£(S3 — Lp/q) for which the "phantom" isotopies are actual isotopies. Received by the editors April 12, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. Key words and phrases. Incompressible surface, 2-bridge link. ©1988 American Mathematical Society 0002-9947/88 $1.00+ $.25 per page 575 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use