Geometry & Topology

Taut ideal triangulations of 3–manifolds

Marc Lackenby

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Abstract

A taut ideal triangulation of a 3–manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2–simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behaviour is very similar to that of a taut foliation. For example, by studying normal surfaces in taut ideal triangulations, we give a new proof of Gabai’s result that the singular genus of a knot in the 3–sphere is equal to its genus.

Article information

Source
Geom. Topol., Volume 4, Number 1 (2000), 369-395.

Dates
Received: 13 April 2000
Revised: 2 November 2000
Accepted: 10 October 2000
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883289

Digital Object Identifier
doi:10.2140/gt.2000.4.369

Mathematical Reviews number (MathSciNet)
MR1790190

Zentralblatt MATH identifier
0958.57019

Subjects
Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
taut ideal triangulation foliation singular genus

Citation

Lackenby, Marc. Taut ideal triangulations of 3–manifolds. Geom. Topol. 4 (2000), no. 1, 369--395. doi:10.2140/gt.2000.4.369. https://projecteuclid.org/euclid.gt/1513883289


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References

  • D Calegari, Foliations transverse to triangulations of 3–manifolds, Comm. Analysis Geom. 8 (2000) 133–158
  • D Gabai, Foliations and the topology of 3–manifolds, J. Differ. Geom. 18 (1983) 445–503
  • D Gabai, Foliations and the topology of 3–manifolds, II, J. Differ. Geom. 26 (1987) 461–478
  • M Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000) 243–282
  • U Oertel, Homology branched surfaces: Thurston's norm on $H_2(M^3)$, from: “Low-Dimensional Topology and Kleinian Groups”, (D B A Epstein editor), London Math. Soc. Lecture Notes, 112 (1986) 253–272
  • L Person, A piecewise linear proof that the singular norm is the Thurston norm, Top. Appl. 51 (1993) 269–289
  • C Petronio, J Porti, Negatively oriented ideal triangulations and a proof of Thurston's hyperbolic Dehn filling theorem, preprint (1999)
  • M Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differ. Geom. 29 (1989) 557–614
  • A Thompson, Thin position and the recognition problem for $S^3$, Math. Res. Lett. 1 (1994) 613–630
  • W Thurston, The Geometry and Topology of 3–manifolds, Princeton University (1978–79)