Algebraic & Geometric Topology

Treewidth, crushing and hyperbolic volume

Clément Maria and Jessica S Purcell

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The treewidth of a 3–manifold triangulation plays an important role in algorithmic 3–manifold theory, and so it is useful to find bounds on the treewidth in terms of other properties of the manifold. We prove that there exists a universal constant c such that any closed hyperbolic 3–manifold admits a triangulation of treewidth at most the product of c and the volume. The converse is not true: we show there exists a sequence of hyperbolic 3–manifolds of bounded treewidth but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects treewidth by at most a constant multiple.

Article information

Algebr. Geom. Topol., Volume 19, Number 5 (2019), 2625-2652.

Received: 8 August 2018
Revised: 21 January 2019
Accepted: 4 February 2019
First available in Project Euclid: 26 October 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 57M15: Relations with graph theory [See also 05Cxx] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds

$3$–manifold triangulation treewidth hyperbolic volume crushing normal surface


Maria, Clément; Purcell, Jessica S. Treewidth, crushing and hyperbolic volume. Algebr. Geom. Topol. 19 (2019), no. 5, 2625--2652. doi:10.2140/agt.2019.19.2625.

Export citation


  • I Agol, Small $3$–manifolds of large genus, Geom. Dedicata 102 (2003) 53–64
  • S Arnborg, D G Corneil, A Proskurowski, Complexity of finding embeddings in a $k$–tree, SIAM J. Algebraic Discrete Methods 8 (1987) 277–284
  • S Arnborg, A Proskurowski, D G Corneil, Forbidden minors characterization of partial $3$–trees, Discrete Math. 80 (1990) 1–19
  • R Benedetti, C Petronio, Lectures on hyperbolic geometry, Springer (1992)
  • D Bienstock, On embedding graphs in trees, J. Combin. Theory Ser. B 49 (1990) 103–136
  • H L Bodlaender, A linear-time algorithm for finding tree-decompositions of small treewidth, SIAM J. Comput. 25 (1996) 1305–1317
  • G Burde, H Zieschang, Knots, De Gruyter Studies in Mathematics 5, de Gruyter, Berlin (1985)
  • B A Burton, Introducing Regina, the $3$–manifold topology software, Experiment. Math. 13 (2004) 267–272
  • B A Burton, A new approach to crushing $3$–manifold triangulations, Discrete Comput. Geom. 52 (2014) 116–139
  • B A Burton, R G Downey, Courcelle's theorem for triangulations, J. Combin. Theory Ser. A 146 (2017) 264–294
  • B A Burton, C Maria, J Spreer, Algorithms and complexity for Turaev–Viro invariants, from “Automata, languages, and programming, I” (M M Halldórsson, K Iwama, N Kobayashi, B Speckmann, editors), Lecture Notes in Comput. Sci. 9134, Springer (2015) 281–293
  • B A Burton, J Spreer, The complexity of detecting taut angle structures on triangulations, from “Proceedings of the Twenty-Fourth Annual ACM–SIAM Symposium on Discrete Algorithms” (S Khanna, editor), SIAM, Philadelphia (2012) 168–183
  • B Courcelle, The monadic second-order logic of graphs, I: Recognizable sets of finite graphs, Inform. and Comput. 85 (1990) 12–75
  • M Culler, N M Dunfield, M Goerner, J R Weeks, SnapPy: a computer program for studying the geometry and topology of $3$–manifolds, software (2016) Available at \setbox0\makeatletter\@url {\unhbox0
  • H Edelsbrunner, Geometry and topology for mesh generation, Cambridge Monographs on Applied and Computational Mathematics 7, Cambridge Univ. Press (2001)
  • W Fenchel, Elementary geometry in hyperbolic space, De Gruyter Studies in Mathematics 11, de Gruyter, Berlin (1989)
  • D Futer, E Kalfagianni, J S Purcell, Dehn filling, volume, and the Jones polynomial, J. Differential Geom. 78 (2008) 429–464
  • F Guéritaud, On canonical triangulations of once-punctured torus bundles and two-bridge link complements, Geom. Topol. 10 (2006) 1239–1284 With an appendix by David Futer
  • K Huszár, J Spreer, U Wagner, On the treewidth of triangulated $3$–manifolds, from “34th International Symposium on Computational Geometry” (B Speckmann, C D Tóth, editors), Leibniz Int. Proc. Inform. 99, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2018) art. id. 46
  • W Jaco, J H Rubinstein, $0$–efficient triangulations of $3$–manifolds, J. Differential Geom. 65 (2003) 61–168
  • W Jaco, J H Rubinstein, Layered-triangulations of $3$–manifolds, preprint (2006)
  • W Jaco, E Sedgwick, Decision problems in the space of Dehn fillings, Topology 42 (2003) 845–906
  • D A Každan, G A Margulis, A proof of Selberg's hypothesis, Mat. Sb. 75 (117) (1968) 163–168 In Russian; translated in Math. USSR-Sb. 4 (1968) 147–152
  • B Kleiner, J Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008) 2587–2855
  • T Kobayashi, Y Rieck, A linear bound on the tetrahedral number of manifolds of bounded volume (after Jørgensen and Thurston), from “Topology and geometry in dimension three” (W Li, L Bartolini, J Johnson, F Luo, R Myers, J H Rubinstein, editors), Contemp. Math. 560, Amer. Math. Soc., Providence, RI (2011) 27–42
  • C Maria, J Spreer, A polynomial time algorithm to compute quantum invariants of $3$–manifolds with bounded first Betti number, from “Proceedings of the Twenty-Eighth Annual ACM–SIAM Symposium on Discrete Algorithms” (P N Klein, editor), SIAM, Philadelphia (2017) 2721–2732
  • G D Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies 78, Princeton Univ. Press (1973)
  • G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)
  • G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)
  • J S Purcell, Hyperbolic knot theory, book project (2018) Available at \setbox0\makeatletter\@url {\unhbox0
  • N Robertson, P D Seymour, Graph minors, II: Algorithmic aspects of tree-width, J. Algorithms 7 (1986) 309–322
  • M Sakuma, J Weeks, Examples of canonical decompositions of hyperbolic link complements, Japan. J. Math. 21 (1995) 393–439
  • A Satyanarayana, L Tung, A characterization of partial $3$–trees, Networks 20 (1990) 299–322
  • P D Seymour, R Thomas, Call routing and the ratcatcher, Combinatorica 14 (1994) 217–241
  • D M Thilikos, M J Serna, H L Bodlaender, Constructive linear time algorithms for small cutwidth and carving-width, from “Algorithms and computation” (D T Lee, S-H Teng, editors), Lecture Notes in Comput. Sci. 1969, Springer (2000) 192–203
  • W P Thurston, The geometry and topology of three-manifolds, lecture notes, Princeton University (1979) Available at \setbox0\makeatletter\@url {\unhbox0