Refined Kirby calculus for integral homology spheres

Abstract

A theorem of Kirby states that two framed links in the $3$–sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on framed links to be able to be replaced with a sequences of algebraically canceling pairs of handle-slides. Then, using the first result, we obtain a refinement of Kirby’s calculus for integral homology spheres which involves only $\pm 1$–framed links with zero linking numbers.