Let M be a compact, connected, irreducible, orientable 3-manifold with boundary a torus, and which is not Seifert fibred. Given a nontrivial simple closed curve r on \(\partial M\) we may construct a closed 3- manifold M(r) by attaching a 2-handle along r and then a 3-handle. The main result of this paper asserts that if M(r) and M(s) are “small” in the sense that they contain no incompressible surfaces and their fundamental groups have no noncyclic representations into PSL(2,\({\mathbb{C}})\) then the minimal geometric intersection number of r and s is at most 1. Hence there are at most three such curves (up to unoriented isotopy).
If M contains no essential torus then its interior has a hyperbolic metric of finite volume, and the methods ofM. Culler andP. B. Shalen [Ann. Math., II. Ser. 117, 109-146 (1983;Zbl 0529.57005)] may be used to show that if M(r) is small then r is a strict boundary slope, i.e. is a boundary component of an essential surface in M which is not a fibre in any fibration of M over the circle. If r or s is a strict boundary slope, or if M contains an essential torus, the remainder of the argument is then reduced to a graph-theoretic analysis of the intersections of certain planar surfaces in M.
This result has already had many striking consequences for knot theory. We shall mention just three. (1) At most one nontrivial surgery on a nontrivial knot in \(S^ 3\) can give a homotopy 3-sphere. (2) Up to unoriented equivalence, there are at most two knots whose complements are of a given homeomorphism type. These are proven in the present paper. Finally, (3)W. Whitten [Topology 26, 41-44 (1987;Zbl 0607.57004)] has recently shown that prime knots with isomorphic groups have homeomorphic complements.

Reviewer: J.Hillman