Algebraic and Geometric Topology 1 (2001), paper no. 7, pages 143-152.

Brunnian links are determined by their complements

Brian Mangum, Theodore Stanford

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.

Keywords. Brunnian, knot, link, link equivalence, link complement

AMS subject classification. Primary: 57M25. Secondary: 57M27.

DOI: 10.2140/agt.2001.1.143

E-print: arXiv:math.GT/9912006

Submitted: 16 November 2000. Revised: 9 February 2001. Accepted: 19 February 2001. Published: 2 March 2001.

Notes on file formats

Brian Mangum, Theodore Stanford
Barnard College, Columbia University
Department of Mathematics
New York, NY 10027, USA
New Mexico State University

Department of Mathematical Sciences
Las Cruces, NM 88003, USA


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