The Classification of Nonsimple Algebraic Tangles

A tangle is a pair (B, T ), where B is a 3-ball, T is a pair of properly embedded arcs. When there is no ambiguity we will simply say that T is a tangle. Let E(T ) = B − IntN(T ) be the exterior of T , usually called the tangle space. T is simple if E(T ) is a simple manifold, that is, it is irreducible, ∂-irreducible, atoroidal, and anannular. By Thurston’s geometrization theorem, simple tangle spaces admit hyperbolic structures with totally geodesic boundary. When embedding the tangle space into S in the natural way, the complement is a handlebody of genus two. Hence, detecting simple tangles is the same as detecting embedded genus two handlebodies with simple complement. In general, it is difficult to determine if a tangle is simple. The first simple tangle (see Figure 1.5(a)) was suggested by Jaco [7, P194], and was verified by Myers [8] with a rather lengthy argument. Due to the nature of the problem, such kind of argument seems unavoidable before a general theory of detecting simple tangles is developed. A similar tangle is the one in Figure 1.5(b), which was proved simple by Ruberman [10]. Another simple tangle is the one in Figure 1.6. Its tangle space was called the tripos manifold, and was proved to be hyperbolic by Thurston [12, Chapter 3]. These have been used by Adams and Reid [1] to discuss quasi-Fuchsian surfaces in knot complements. It seems that these are essentially the only tangles which were known to be simple. A tangle is called an algebraic tangle if it can be obtained by summing up finitely many rational tangles in various ways. An algebraic tangle is a Montesinos tangle if all the gluing disks are disjoint from each other. By analyzing incompressible annuli in exteriors of algebraic tangles, we will be able to prove Theorem 4.9, which completely classifies all nonsimple algebraic tangles. The paper is organized as follows. In Section 1 we will state the classification theorems, prove corollaries, and show some examples. Section 2 is to classify all marked tangles containing either a monogon or a bigon. The results will be used in Section 3 to give the proofs of Theorem 3.6. In Section 4 we will determine all marked algebraic tangles which are ∆-annular (see Section 1 for definitions), and use the result to prove the classification theorem of nonsimple algebraic tangles. Mathematics Subject Classification: 57N10, 57M25, 57M50. Partially supported by NSF grant DMS 9102633