Brunnian links are determined by their complements

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

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M27

References

Forward citations

Publication

Received: 16 November 2000

Accepted: 28 February 2001

Published: 2 March 2001

Authors

Brian S Mangum
Barnard College Columbia University Department of Mathematics New York NY 10027 USA

Theodore Stanford
New Mexico State University Department of Mathematical Sciences Las Cruces NM 88003 USA

Volume 1, issue 1 (2001)

Brian S Mangum and Theodore Stanford