The Classification of Dehn Surgeries on 2-bridge Knots

We will determine whether a given surgery on a 2-bridge knot is reducible, toroidal, Seifert bered, or hyperbolic. In [Th1] Thurston showed that if K is a hyperbolic knot, then all but nitely many surgeries on K are hyperbolic. In particular, for the Figure 8 knot, it was shown that exactly 9 nontrivial surgeries are non-hyperbolic. Let Kp=q be a 2-bridge knot associated to the rational number p=q. When p 1 mod q,K is a torus knot, on which the surgeries are well understood. By [HT], all other 2-bridge knots are hyperbolic, admitting no reducible surgeries. Moreover, Kp=q admits a toroidal surgery if and only if p=q = [r1; r2] = 1=(r1 1=r2) for some integers r1; r2. See Lemma 8 below for a complete list of all toroidal surgeries. The Geometrization Conjecture [Th2] asserts that if a closed orientable 3-manifold is irreducible and atoroidal, then it is either a hyperbolic manifold, or a Seifert bered space whose orbifold is a 2-sphere with at most three cone points, called a small Seifert bered space. The conjecture has been proved for two large classes of manifolds: the Haken manifolds [Th2], and those admitting an orientation preserving periodic map with nonempty xed point set [Th3,Ho,KOS,Zh]. It can be shown that surgery on a 2-bridge knot yields a manifold which admits such a periodic map, so it has a geometric decomposition. Our main result will classify all surgeries on 2-bridge knots according to whether they are reducible, toroidal, Seifert bered, or hyperbolic manifolds. 1991 Mathematics Subject Classi cation. 57N10, 57M25, 57M50.