## Algebraic & Geometric Topology

### Brunnian links are determined by their complements

#### 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

#### 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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