Toroidally alternating knots and links

Abstract

THERE have been various generalizations of the class of alternating links in S3, specifically alternative, pseudo-alternating, homogeneous, adequate, augmented alternating and almost alternating links. These generalizations originated out of attempts to extend results known for alternating links to broader classes of links. In this paper, we extend the class of alternating links to a new set, which we call toroidally alternating links. This set of links will be particularly broad, containing within it the set of alternating links, the set of almost alternating links, the subset of augmented alternating links with a single augmenting component, and a sub-class of the set of arborescent links, including all Montesinos links. Surprisingly, if the tables of prime knots through eleven crossings and prime nonsplittable links through ten crossings appearing in [6] are examined, all but three of the knots and two of the links can be shown to be toroidally alternating. Let T be a torus embedded in an orientable 3-manifold M. Let L be a link in M that can be isotoped into a neighborhood T x I of T. Suppose that if T x I is retracted onto T, L projects to a connected 4-valent graph on T such that if one keeps track of the crossings, they alternate between over and under as the components of the link are traversed, when viewed from one side of T. In addition, assume that every nontrivial curve on T intersects the projection of L onto T. Then L is said to be toroidally alternating with respect to T. In the case of a manifold with a genus one Heegaard splitting, there is a unique torus up to isotopy that splits the manifold into two solid tori. (See [3].) Hence, we can define a toroidally alternating link in these manifolds to be a link that is toroidally alternating with respect to this particular torus. We will prove that a toroidally alternating knot in S3 contains no closed incompressible meridianally incompressible surfaces in its complement. In particular, this will mean that a prime nontrivial toroidally alternating knot in S3 is either a torus knot or it is hyperbolic. This generalizes a result that was proved for alternating links in [ 1 l] and almost alternating knots in [2]. In fact, we will prove that this is also the case when S3 is replaced by a lens space L(p, q) where p is odd. In the case p is even, toroidally alternating knots can have incompressible meridianally incompressible surfaces in their complement. However, we will show that if K is a nontrivial prime non-torus toroidally alternating knot in L(p, q), then L(p, q) K is hyperbolic, except for the lens spaces homeomorphic to L(2, 1) and L(4k, 2k l), where hyperbolicity will depend on the particular choice of a toroidally alternating knot. We will