Dehn surgeries on 2-bridge links which yield reducible 3-manifolds.pdf

a r X i v : m a t h / 0 5 1 2 1 1 6 v 1 [ m a t h .G T ] 6 D e c 2 0 0 5 DEHN SURGERIES ON 2-BRIDGE LINKS WHICH YIELD REDUCIBLE 3-MANIFOLDS HIROSHI GODA, CHUICHIRO HAYASHI AND HYUN-JONG SONG 1. Introduction No surgery on a non-torus 2-bridge knot yields a reducible 3-manifold as shown in The- orem 2(a) in [7] by A. Hatcher and W. Thurston. Dehn surgeries on 2-bridge knots are already well studied by M. Brittenham and Y.-Q. Wu in [2]. See also [11]. On 2-bridge links of 2-components, Wu showed in [13, Theorem 5.1 and Remark 5.5] the following theorem. The universal covering space of a laminar 3-manifoldM is the Euclidean 3-space R3, and then M is not reducible by [4]. Theorem 1.1. ([13]) A non-trivial Dehn surgery on a 2-bridge link yields a laminar 3- manifold except for the 2-bridge links L([r, s]) with [r, s] = 1/(r + 1/s). For them with r, s 6= ±1, we obtain laminar 3-manifolds if the two surgery slopes are both non-integral. In this paper we study Dehn surgery on 2-bridge links L([r, s]). Note that L([r, s]) is a link of 2-components if and only if both r and s are odd integers. Since L([r, s]) ～= L([?s,?r]) and L([?r,?s]) ～= L([s, r]) is the mirror image of L([r, s]), it is enough to consider L([r, s]) with r 0. When |r| = 1 or |s| = 1, L([r, s]) is a (2, k)-torus link for some even integer k. Thus we consider the case where r ≥ 3 and either s ≥ 3 or s ≤ ?3. Figure 1. Every component of a 2-bridge link is the trivial knot. We coordinate slopes on the toral boundaries of the link exterior so that the slope in Figure 1 is +3 rather than ?3. Let This work was supported by Joint Research Project ‘Geometric and Algebraic Aspects of Knot Theory’, under the Japan-Korea Basic Scientific Cooperation Program by KOSEF and JSPS. The authors would like to thank Professor Hitoshi Murakami for giving us this opportunity. The first and second authors are partially supported by Grant-in-Aid for Scientific Research, (No.and No.respectively), Ministry