Special positions for essential Tori in link complements

Abstract

2-spheres, since the latter cannot be knotted. The first major attempt to understand embedded tori in link complements was a groundbreaking paper by Schubert [6]. The seminal role which is played in the topology and geometry of link complements by embedded tori was later underscored in the important work of Jaco and Shalen [3], Johansson [4] and Thurston [7], who showed that if M3 is a 3-manifold, then there is a finite collection R of essential, non-peripheral tori T,,. . .,T, in M3 such that each component of M3 split open along the tori in 0 is either Seifert-fibered or hyperbolic. Our goal in this paper is to apply the techniques of [2] to the study of essential tori in link complements. Let II be a link type in S3, with representative L. A torus T in S3-L is essential if it is incompressible, and peripheral if it is parallel to the boundary of a tubular neighborhood of L. A link type E. is simple if every essential torus in its complement is peripheral, otherwise non-simple. Satellite links (defined below) are a special case of non-simple links. To describe our results, assume that L is a closed n-braid representative of 1, with braid axis A. The axis A is unknotted, so S3-A is fibered by open discs {H,; 8 E [0,2rc]}. It will be convenient to think of A as the Z-axis in R3, and the fibers H, as half-planes at polar angle 8. Then A and the half-planes H, serve as a “coordinate system” in R3 which can be used to describe both L and T. We call our canonical embeddings types 0, 1 and k, where in the latter case k is an integer r 2. Type 0 will be familiar to most readers, but types 1 and k do not appear to have been noticed before this as general phenomena: Type 0. The torus T is the boundary of a (possibly knotted) solid torus V in S3, whose core is a closed braid with axis A. The link L is also a closed braid with respect to A, part of it being inside V and part of it (possibly empty) outside. The torus T is transverse to every fiber H, in the fibration of S3-A, and intersects each fiber in a meridian of V. It is foliated by