Algebraic & Geometric Topology

An alternative approach to hyperbolic structures on link complements

Morwen Thistlethwaite and Anastasiia Tsvietkova

Full-text: Open access


An alternative method is described for determining the hyperbolic structure on a link complement, and some of its elementary consequences are examined. The method is particularly suited to alternating links.

Article information

Algebr. Geom. Topol., Volume 14, Number 3 (2014), 1307-1337.

Received: 1 August 2011
Revised: 9 September 2013
Accepted: 29 September 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds

classical link hyperbolic structure tangle


Thistlethwaite, Morwen; Tsvietkova, Anastasiia. An alternative approach to hyperbolic structures on link complements. Algebr. Geom. Topol. 14 (2014), no. 3, 1307--1337. doi:10.2140/agt.2014.14.1307.

Export citation


  • C C Adams, Thrice-punctured spheres in hyperbolic $3$–manifolds, Trans. Amer. Math. Soc. 287 (1985) 645–656
  • C C Adams, Waist size for cusps in hyperbolic $3$–manifolds, Topology 41 (2002) 257–270
  • C C Adams, Noncompact Fuchsian and quasi-Fuchsian surfaces in hyperbolic $3$–manifolds, Algebr. Geom. Topol. 7 (2007) 565–582
  • I R Aitchison, E Lumsden, J H Rubinstein, Cusp structures of alternating links, Invent. Math. 109 (1992) 473–494
  • L Bers, Quasiconformal mappings and Teichmüller's theorem, from: “Analytic functions”, Princeton Mathematical Series 24, Princeton Univ. Press (1960) 89–119
  • R D Canary, D McCullough, Homotopy equivalences of $3$–manifolds and deformation theory of Kleinian groups, Mem. Amer. Math. Soc. 812, Amer. Math. Soc. (2004)
  • J H Conway, An enumeration of knots and links, and some of their algebraic properties, from: “Computational problems in abstract algebra”, (J Leech, editor), Pergamon, Oxford (1970) 329–358
  • D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80
  • S Francaviglia, Hyperbolic volume of representations of fundamental groups of cusped $3$–manifolds, Int. Math. Res. Not. 2004 (2004) 425–459
  • D Futer, E Kalfagianni, J Purcell, Quasifuchsian state surfaces, to appear in Trans. Amer. Math. Soc. (2012)
  • W B R Lickorish, Prime knots and tangles, Trans. Amer. Math. Soc. 267 (1981) 321–332
  • A Marden, Geometrically finite Kleinian groups and their deformation spaces, from: “Discrete groups and automorphic functions”, (W J Harvey, editor), Academic Press, London (1977) 259–293
  • W Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984) 37–44
  • W Menasco, M Thistlethwaite, The classification of alternating links, Ann. of Math. 138 (1993) 113–171
  • J W Morgan, On Thurston's uniformization theorem for three-dimensional manifolds, from: “The Smith conjecture”, (J W Morgan, H Bass, editors), Pure Appl. Math. 112, Academic Press, Orlando, FL (1984) 37–125
  • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Houston, TX (1990)
  • D Ruberman, Mutation and volumes of knots in $S\sp 3$, Invent. Math. 90 (1987) 189–215
  • M Sakuma, J Weeks, Examples of canonical decompositions of hyperbolic link complements, Japan. J. Math. 21 (1995) 393–439