If and are two Brunnian links with all pairwise linking numbers , then we show that and are equivalent if and only if they have homeomorphic complements. In particular, this holds for all Brunnian links with at least three components. If is a Brunnian link with all pairwise linking numbers , and the complement of is homeomorphic to the complement of , then we show that may be obtained from by a sequence of twists around unknotted components. Finally, we show that for any positive integer , an algorithm for detecting an –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.
"Brunnian links are determined by their complements." Algebr. Geom. Topol. 1 (1) 143 - 152, 2001. https://doi.org/10.2140/agt.2001.1.143