Algebraic & Geometric Topology

Volumes of highly twisted knots and links

Jessica Purcell

Full-text: Open access


We show that for a large class of knots and links with complements in S3 admitting a hyperbolic structure, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a knot or link admits a prime, twist reduced diagram with at least 2 twist regions and at least C crossings per twist region, then the link complement is hyperbolic with volume bounded below by 3.3515 times the number of twist regions in the diagram. C is at most 113.

Article information

Algebr. Geom. Topol., Volume 7, Number 1 (2007), 93-108.

Received: 21 April 2006
Revised: 3 January 2007
Accepted: 3 January 2007
First available in Project Euclid: 20 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

hyperbolic knot complements hyperbolic link complements volume cone manifolds


Purcell, Jessica. Volumes of highly twisted knots and links. Algebr. Geom. Topol. 7 (2007), no. 1, 93--108. doi:10.2140/agt.2007.7.93.

Export citation


  • C C Adams, Augmented alternating link complements are hyperbolic, from: “Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984)”, London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press, Cambridge (1986) 115–130
  • I Agol, N Dunfield, P Storm, W P Thurston, Lower bounds on volumes of hyperbolic Haken 3-manifolds
  • K B ör öczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978) 243–261
  • C Cao, G R Meyerhoff, The orientable cusped hyperbolic $3$-manifolds of minimum volume, Invent. Math. 146 (2001) 451–478
  • D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan, Tokyo (2000) With a postface by Sadayoshi Kojima
  • D Futer, J S Purcell, Links with no exceptional surgeries
  • C D Hodgson, Degeneration and regeneration of geometric structures on $3$-manifolds, PhD thesis, Princeton Univ. (1986)
  • C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1–59
  • C D Hodgson, S P Kerckhoff, Universal bounds for hyperbolic Dehn surgery, Ann. of Math. $(2)$ 162 (2005) 367–421
  • M Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. $(3)$ 88 (2004) 204–224 With an appendix by Ian Agol and Dylan Thurston
  • C J Leininger, Small curvature surfaces in hyperbolic 3-manifolds, J. Knot Theory Ramifications 15 (2006) 379–411
  • J S Purcell, Cusp shapes under cone deformation
  • J S Purcell, Cusp Shapes of Hyperbolic Link Complements and Dehn Filling, PhD thesis, Stanford University (2004)
  • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Berkeley, CA (1976)
  • W P Thurston, The Geometry and Topology of Three-Manifolds, Princeton Univ. Math. Dept. Notes (1979)
  • J R Weeks, Snappea