Algebraic & Geometric Topology

The length of unknotting tunnels

Daryl Cooper, Marc Lackenby, and Jessica S Purcell

Full-text: Open access


We show there exist tunnel number one hyperbolic 3–manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.

Article information

Algebr. Geom. Topol., Volume 10, Number 2 (2010), 637-661.

Received: 11 August 2009
Accepted: 13 January 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds

unknotting tunnel hyperbolic $3$–manifold geodesic


Cooper, Daryl; Lackenby, Marc; Purcell, Jessica S. The length of unknotting tunnels. Algebr. Geom. Topol. 10 (2010), no. 2, 637--661. doi:10.2140/agt.2010.10.637.

Export citation


  • C Adams, Unknotting tunnels in hyperbolic $3$–manifolds, Math. Ann. 302 (1995) 177–195
  • C Adams, A W Reid, Unknotting tunnels in two-bridge knot and link complements, Comment. Math. Helv. 71 (1996) 617–627
  • H Akiyoshi, Y Nakagawa, M Sakuma, Shortest vertical geodesics of manifolds obtained by hyperbolic Dehn surgery on the Whitehead link, from: “KNOTS '96 (Tokyo)”, (S Suzuki, editor), World Sci. Publ., River Edge, NJ (1997) 433–448
  • L Bers, On moduli of Kleinian groups, Uspehi Mat. Nauk 29 (1974) 86–102 Translated from the English by M E Novodvorskiĭ, Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I
  • B H Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993) 245–317
  • R D Canary, M Culler, S Hersonsky, P B Shalen, Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups, J. Differential Geom. 64 (2003) 57–109
  • M Culler, P B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. $(2)$ 117 (1983) 109–146
  • D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80
  • A Fathi, F Laudenbach, V Poenaru, editors, Travaux de Thurston sur les surfaces, Astérisque 66–67, Soc. Math. France, Paris (1979) Séminaire Orsay, With an English summary
  • D J Heath, H-J Song, Unknotting tunnels for $P(-2,3,7)$, J. Knot Theory Ramifications 14 (2005) 1077–1085
  • M Lackenby, J S Purcell, Geodesic arcs in compression bodies, in preparation
  • C Lecuire, An extension of the Masur domain, Preprint
  • A Marden, The geometry of finitely generated kleinian groups, Ann. of Math. $(2)$ 99 (1974) 383–462
  • A Marden, Outer circles. An introduction to hyperbolic $3$–manifolds, Cambridge Univ. Press (2007)
  • B Maskit, Kleinian groups, Grund. der Math. Wissenschaften 287, Springer, Berlin (1988)
  • C McMullen, Cusps are dense, Ann. of Math. $(2)$ 133 (1991) 217–247
  • J W Morgan, H Bass, editors, The Smith conjecture, Pure and Applied Math. 112, Academic Press, Orlando, FL (1984) Papers presented at the symposium held at Columbia University, New York, 1979
  • M Sakuma, J Weeks, Examples of canonical decompositions of hyperbolic link complements, Japan. J. Math. $($N.S.$)$ 21 (1995) 393–439
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417–431
  • J Weeks, SnapPea Available at \setbox0\makeatletter\@url {\unhbox0