#### Volume 1, issue 1 (2001)

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Brunnian links are determined by their complements

### Brian S Mangum and Theodore Stanford

Algebraic & Geometric Topology 1 (2001) 143–152
 arXiv: math.GT/9912006
##### Abstract

If ${L}_{1}$ and ${L}_{2}$ are two Brunnian links with all pairwise linking numbers $0$, then we show that ${L}_{1}$ and ${L}_{2}$ are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If ${L}_{1}$ is a Brunnian link with all pairwise linking numbers $0$, and the complement of ${L}_{2}$ is homeomorphic to the complement of ${L}_{1}$, then we show that ${L}_{2}$ may be obtained from ${L}_{1}$ 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.