A simple proof of the fundamental theorem of Kirby calculus on links

Abstract

Abstract. In this paper, we relate surgeries on links and Heegaard decompositions, relate framed links and surface mapping classes, and give a simple proof of the fundamental theorem of Kirby calculus on links by the presentation of the surface mapping class groups.Surgery on links, a beautiful way to describe 3-manifolds, was discovered independently by Wallace [Wa] and Lickorish [Li] in the early sixties. It has become even more attractive since Kirby found a criterion-so called Kirby calculus today-to tell how two such surgeries which define the same 3-manifold are related by some combinatorial moves. In Kirby's original proof Cerf theory played a central role which was not easily understood. A more intuitive proof was suggested by Hatcher-Thurston-Wajnryb's explicit presentation [W] of the surface mapping class groups. In this paper, we are going to give a new proof of the fundamental theorem of Kirby calculus using this idea. Instead of proving Kirby's original version, we will prove directly the simpler version of FennRourke [FR]: Theorem 1.1 (Kirby-Fenn-Rourke). Two integer framed links determine the same 3-manifold if and only if they are related by K±x-moves. Moreover, if all components of these two links are unknotted, then all K±x-moves can be chosen such that all links involved in the process have unknotted components.A one-page proof of the equivalence of these two versions can be found in the paper [FR].A K-move, as pictured in Figure 1.1, is defined to be an elimination of a ±1-labeled unknotted component of the link by doing the corresponding surgery. And a K~x -move is the reverse process of such a K-move.A nice thing in our proof is that one actually can see how Heegaard decomposition-another important way to describe 3-manifold-and surgery on links can be transformed to each other. In addition, if two framed links determine the same 3-manifold our proof gives an algorithm for relating them by K±x-moves.