Algebraic & Geometric Topology

Brunnian links are determined by their complements

Brian S Mangum and Theodore Stanford

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Abstract

If L1 and L2 are two Brunnian links with all pairwise linking numbers 0, then we show that L1 and L2 are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If L1 is a Brunnian link with all pairwise linking numbers 0, and the complement of L2 is homeomorphic to the complement of L1, then we show that L2 may be obtained from L1 by a sequence of twists around unknotted components. Finally, we show that for any positive integer n, an algorithm for detecting an n–component unlink leads immediately to an algorithm for detecting an unlink of any number of components. This algorithmic generalization is conceptually simple, but probably computationally impractical.

Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 143-152.

Dates
Received: 16 November 2000
Accepted: 28 February 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882587

Digital Object Identifier
doi:10.2140/agt.2001.1.143

Mathematical Reviews number (MathSciNet)
MR1823496

Zentralblatt MATH identifier
0970.57002

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

Keywords
Brunnian knot link link equivalence link complement

Citation

Mangum, Brian S; Stanford, Theodore. Brunnian links are determined by their complements. Algebr. Geom. Topol. 1 (2001), no. 1, 143--152. doi:10.2140/agt.2001.1.143. https://projecteuclid.org/euclid.agt/1513882587


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