Vol. 99, No. 2, 1982

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On the boundary curves of incompressible surfaces

Allen E. Hatcher

Vol. 99 (1982), No. 2, 373–377

Let K be a knot in S3, and consider incompressible (in the stronger sense of π1-injective), -incompressible surfaces S in the exterior of K. A question which has been around for some time is whether the boundary-slope function SmS∕lS, where mS and lS are the numbers of times each circle of ∂S wraps around K meridionally and longitudinally, takes on only finitely many values (for fixed K). This is known to be true for certain knots: torus knots, the figure-eight knot [4], 2-bridge knots [2], and alternating knots [3]. In this paper an affirmative answer is given not just for knot exteriors, but for all compact orientable irreducible 3-manifolds M with ∂M a torus. Further, we give a natural generalization to the case when ∂M is a union of tori.

Mathematical Subject Classification 2000
Primary: 57N10
Received: 15 October 1980
Revised: 2 December 1980
Published: 1 April 1982
Allen E. Hatcher
Mathematics Department
Cornell University
Ithaca NY 14853
United States