#### 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

Email: mangum@math.columbia.edu, stanford@nmsu.edu

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