Algebraic & Geometric Topology

Brunnian links are determined by their complements

Brian S Mangum and Theodore Stanford

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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.

Article information

Algebr. Geom. Topol., Volume 1, Number 1 (2001), 143-152.

Received: 16 November 2000
Accepted: 28 February 2001
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

Brunnian knot link link equivalence link complement


Mangum, Brian S; Stanford, Theodore. Brunnian links are determined by their complements. Algebr. Geom. Topol. 1 (2001), no. 1, 143--152. doi:10.2140/agt.2001.1.143.

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  • John Berge. The knots in ${D}\sp 2\times {S}\sp 1$ which have nontrivial Dehn surgeries that yield ${D}\sp 2\times {S}\sp 1$. Topology Appl., 38(1):1–19, 1991.
  • Gerhard Burde and Kunio Murasugi. Links and Seifert fiber spaces. Duke Math. J., 37:89–93, 1970.
  • Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen. Dehn surgery on knots. Ann. of Math. (2), 125(2):237–300, 1987.
  • Hans Debrunner. Links of Brunnian type. Duke Math. J., 28:17–23, 1961.
  • David Gabai. Foliations and the topology of $3$-manifolds. II. J. Differential Geom., 26(3):461–478, 1987.
  • C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371–415, 1989.
  • Yves Mathieu. Unknotting, knotting by twists on disks and property $({\rm {p}})$ for knots in ${S}\sp 3$. In Knots 90 (Osaka, 1990), pages 93–102. de Gruyter, Berlin, 1992.
  • Dale Rolfsen. Knots and links. Publish or Perish Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7.
  • Joachim H. Rubinstein. An algorithm to recognize the $3$-sphere. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 601–611, Basel, 1995. Birkhäuser.
  • Abigail Thompson. Thin position and the recognition problem for ${S}\sp 3$. Math. Res. Lett., 1(5):613–630, 1994.
  • Ying Qing Wu. Incompressibility of surfaces in surgered $3$-manifolds. Topology, 31(2):271–279, 1992.