Algebraic & Geometric Topology

Meridional almost normal surfaces in knot complements

Robin Wilson

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Suppose K is a knot in a closed 3–manifold M such that MN(K) is irreducible. We show that for any integer n there exists a triangulation of MN(K) such that any weakly incompressible bridge surface for K of n bridges or fewer is isotopic to an almost normal bridge surface.

Article information

Algebr. Geom. Topol., Volume 8, Number 3 (2008), 1717-1740.

Received: 6 October 2007
Accepted: 1 September 2008
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 57M99: None of the above, but in this section

normal surface Heegaard surface bridge position strongly irreducible weakly incompressible


Wilson, Robin. Meridional almost normal surfaces in knot complements. Algebr. Geom. Topol. 8 (2008), no. 3, 1717--1740. doi:10.2140/agt.2008.8.1717.

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  • D Bachman, Heegaard splittings with boundary and almost normal surfaces, Topology Appl. 116 (2001) 153–184
  • D Bachman, Thin position with respect to a Heegaard surface, preprint
  • A J Casson, C M Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283
  • A Coward, Algorithmically detecting the bridge number of hyperbolic knots (2006)
  • C Frohman, The topological uniqueness of triply periodic minimal surfaces in ${\bf R}\sp 3$, J. Differential Geom. 31 (1990) 277–283
  • D Gabai, Foliations and the topology of $3$-manifolds. III, J. Differential Geom. 26 (1987) 479–536
  • W Haken, Theorie der Normalflächen, Acta Math. 105 (1961) 245–375
  • W Jaco, D Letscher, H Rubinstein, One vertex, ideal, and efficient triangulations of 3–manifolds, in preparation
  • W Jaco, J H Rubinstein, $0$–efficient triangulations of 3–manifolds, J. Differential Geom. 65 (2003) 61–168
  • S King, Almost normal Heegaard surfaces (2003)
  • 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
  • M Scharlemann, A Thompson, Heegaard splittings of $({\rm surface})\times I$ are standard, Math. Ann. 295 (1993) 549–564
  • M Scharlemann, M Tomova, Uniqueness of bridge surfaces for 2–bridge knots, Math. Proc. Cambridge Philos. Soc. 144 (2008) 639–650
  • M Stocking, Almost normal surfaces in $3$–manifolds, Trans. Amer. Math. Soc. 352 (2000) 171–207
  • A Thompson, Thin position and the recognition problem for $S\sp 3$, Math. Res. Lett. 1 (1994) 613–630
  • M Tomova, Multiple bridge surfaces restrict knot distance, Algebr. Geom. Topol. 7 (2007) 957–1006