Geometry & Topology

Refined Kirby calculus for integral homology spheres

Kazuo Habiro

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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 ±1–framed links with zero linking numbers.

Article information

Geom. Topol., Volume 10, Number 3 (2006), 1285-1317.

Received: 20 December 2005
Accepted: 20 June 2006
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

Kirby calculus framed link surgery handle-slide integral homology sphere band-slide Hoste move


Habiro, Kazuo. Refined Kirby calculus for integral homology spheres. Geom. Topol. 10 (2006), no. 3, 1285--1317. doi:10.2140/gt.2006.10.1285.

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