Algebraic & Geometric Topology

On Kirby calculus for null-homotopic framed links in $3$–manifolds

Kazuo Habiro and Tamara Widmer

Full-text: Open access


Kirby proved that two framed links in S3 give orientation-preserving homeomorphic results of surgery if and only if these two links are related by a sequence of two kinds of moves called stabilizations and handle-slides. Fenn and Rourke gave a necessary and sufficient condition for two framed links in a closed, oriented 3–manifold to be related by a finite sequence of these moves.

The purpose of this paper is twofold. We first give a generalization of Fenn and Rourke’s result to 3–manifolds with boundary. Then we apply this result to the case of framed links whose components are null-homotopic in the 3–manifold.

Article information

Algebr. Geom. Topol., Volume 14, Number 1 (2014), 115-134.

Received: 23 February 2013
Revised: 19 June 2013
Accepted: 21 June 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

$3$–manifold framed link surgery Kirby calculus null-homotopic link


Habiro, Kazuo; Widmer, Tamara. On Kirby calculus for null-homotopic framed links in $3$–manifolds. Algebr. Geom. Topol. 14 (2014), no. 1, 115--134. doi:10.2140/agt.2014.14.115.

Export citation


  • A Adem, R J Milgram, Cohomology of finite groups, Grundl. Math. Wissen. 309, Springer, Berlin (1994)
  • J Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, IHÉS Publ. Math. 39 (1970) 5–173
  • T D Cochran, A Gerges, K Orr, Dehn surgery equivalence relations on $3$–manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 97–127
  • C J Earle, J Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967) 557–559
  • R Fenn, C Rourke, On Kirby's calculus of links, Topology 18 (1979) 1–15
  • S Garoufalidis, A Kricker, A surgery view of boundary links, Math. Ann. 327 (2003) 103–115
  • R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)
  • K Habiro, Refined Kirby calculus for integral homology spheres, Geom. Topol. 10 (2006) 1285–1317
  • K Habiro, T Widmer, Kirby calculus for null-homologous framed links in $3$–manifolds, in preparation
  • R Kirby, A calculus for framed links in $S^{3}$, Invent. Math. 45 (1978) 35–56
  • J Roberts, Kirby calculus in manifolds with boundary, Turkish J. Math. 21 (1997) 111–117