Algebraic & Geometric Topology

Ideal triangulations of 3-manifolds II; taut and angle structures

Ensil Kang and J Hyam Rubinstein

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This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3–manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program SNAPPEA of Weeks, and the program SNAP of Coulson, Goodman, Hodgson and Neumann. Casson has also written a program to find hyperbolic structures on such 3–manifolds, by solving Thurston’s hyperbolic gluing equations for ideal triangulations. In this second paper, we study the question of when a taut ideal triangulation of an irreducible atoroidal 3–manifold admits a family of angle structures. We find a combinatorial obstruction, which gives a necessary and sufficient condition for the existence of angle structures for taut triangulations. The hope is that this result can be further developed to give a proof of the existence of ideal triangulations admitting (complete) hyperbolic metrics. Our main result answers a question of Lackenby. We give simple examples of taut ideal triangulations which do not admit an angle structure. Also we show that for ‘layered’ ideal triangulations of once-punctured torus bundles over the circle, that if the manodromy is pseudo Anosov, then the triangulation admits angle structures if and only if there are no edges of degree 2. Layered triangulations are generalizations of Thurston’s famous triangulation of the Figure–8 knot space. Note that existence of an angle structure easily implies that the 3–manifold has a CAT(0) or relatively word hyperbolic fundamental

Article information

Algebr. Geom. Topol., Volume 5, Number 4 (2005), 1505-1533.

Received: 19 May 2005
Accepted: 13 June 2005
First available in Project Euclid: 20 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: 57N10: Topology of general 3-manifolds [See also 57Mxx]

normal surfaces 3–manifolds ideal triangulations taut angle structures


Kang, Ensil; Rubinstein, J Hyam. Ideal triangulations of 3-manifolds II; taut and angle structures. Algebr. Geom. Topol. 5 (2005), no. 4, 1505--1533. doi:10.2140/agt.2005.5.1505.

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  • I Agol, Bounds on exceptional Dehn filling, \gtref4200015431449
  • B Burton, E Kang, J H Rubinstein, Ideal triangulations of $3$–manifolds III; Taut structures in low census manifolds, in preparation
  • P J Callahan, M V Hildebrand, J R Weeks, A census of cusped hyperbolic $3$–manifolds, Math. Comp. 68 (1999) 321–332
  • D Coulson, O A Goodman, C D Hodgson, W D Neumann, Computing arithmetic invariants of $3$–manifolds, Experiment. Math. 9 (2000) 127–152
  • M Freedman, J Hass, P Scott, Least area incompressible surfaces in $3$–manifolds, Invent. Math. 71 (1983) 609–642
  • F Gueritaud, On canonical triangulations of the mapping tori over the punctured torus.
  • W Haken, Theorie der Normalflächen, Acta Math. 105 (1961) 245–375
  • W Haken, Some results on surfaces in $3$–manifolds, from: “Studies in Modern Topology”, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.) (1968) 39–98
  • J Hempel, $3$–Manifolds, Princeton University Press, Princeton, N. J. (1976)
  • W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society, Providence, RI (1980)
  • W Jaco, J H Rubinstein, $0$–efficient triangulations of $3$–manifolds, J. Differential Geom. 65 (2003) 61–168
  • W Jaco, J H Rubinstein, $1$–efficient triangulations of $3$–manifolds, in preparation
  • E Kang, Normal surfaces in knot complements, PhD thesis, University of Connecticut (1999)
  • E Kang, Normal surfaces in non-compact 3–manifolds, J. Aust. Math. Soc. 78 (2005) 305–321
  • E Kang, J H Rubinstein, Ideal triangulations of 3–manifolds I; spun normal surface theory, from: “Proceedings of the Casson Fest (Arkansas and Texas 2003)”, \gtmref7200410235265
  • M Lackenby, Taut ideal triangulations of $3$–manifolds, \gtref4200012369395
  • M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
  • E E Moise, Affine structures in $3$–manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952) 96–114
  • J H Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for $3$–dimensional manifolds, from: “Geometric topology (Athens, GA, 1993)”, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI (1997) 1–20
  • J Simon, Compactification of covering spaces of compact $3$–manifolds, Michigan Math. J. 23 (1976) 245–256 (1977)
  • M Stocking, Almost normal surfaces in $3$–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171–207
  • W Thurston, Three-dimensional geometry and topology, Vol 1, (Silvio Levy, editor), Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ (1997)
  • S Tillmann, Degenerations and normal surface theory, PhD thesis, University of Melbourne (2002)
  • J L Tollefson, Normal surface $Q$–theory, Pacific J. Math. 183 (1998) 359–374