Algebraic & Geometric Topology

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

Kazuo Habiro and Tamara Widmer

Full-text: Open access

Abstract

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

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

Dates
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
https://projecteuclid.org/euclid.agt/1513715796

Digital Object Identifier
doi:10.2140/agt.2014.14.115

Mathematical Reviews number (MathSciNet)
MR3158755

Zentralblatt MATH identifier
1278.57007

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

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

Citation

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. https://projecteuclid.org/euclid.agt/1513715796


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