2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221690A 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 +/-1-framed links with zero linking numbers.This is the version published by Geometry & Topology on 18 September 2006Geometric TopologyQuantum Algebra57M25, 57M27text