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

Habiro, Kazuo; Widmer, Tamara

doi:10.2140/agt.2014.14.115

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On Kirby calculus for null-homotopic framed links in

$3$ –manifolds

Kazuo Habiro and Tamara Widmer

Algebraic & Geometric Topology 14 (2014) 115–134

DOI: 10.2140/agt.2014.14.115

Abstract

Kirby proved that two framed links in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math>$ 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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –manifolds with boundary. Then we apply this result to the case of framed links whose components are null-homotopic in the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –manifold.

Keywords

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

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

References

Publication

Received: 23 February 2013

Revised: 19 June 2013

Accepted: 21 June 2013

Preview posted: 21 November 2013

Published: 9 January 2014

Authors

Kazuo Habiro
Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan

Tamara Widmer
Institut für Mathematik Universität Zürich Winterthurerstr. 190 CH-8057 Zürich Switzerland

Volume 14, issue 1 (2014)