On the boundary curves of incompressible surfaces

Abstract

Let K be a knot in S 3 , and consider incompressible (in the stronger sense of πi-injective), d-incompressible surfaces S in the exterior of K.A question which has been around for some time is whether the boundary-slope function SH-» msISs, where m s and S s are the numbers of times each circle of dS 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 dM a torus.Further, we give a natural generalization to the case when dM is a union of tori.

Locations

• Pacific Journal of Mathematics - View - PDF