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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Brunnian links are determined by their complements

Brian S Mangum and Theodore Stanford

Algebraic & Geometric Topology 1 (2001) 143–152

arXiv: math.GT/9912006


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.

Brunnian, knot, link, link equivalence, link complement
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27
Forward citations
Received: 16 November 2000
Accepted: 28 February 2001
Published: 2 March 2001
Brian S Mangum
Barnard College
Columbia University
Department of Mathematics
New York NY 10027
Theodore Stanford
New Mexico State University
Department of Mathematical Sciences
Las Cruces NM 88003