Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Core curves of triangulated solid tori

Author: Marc Lackenby
Journal: Trans. Amer. Math. Soc. 366 (2014), 6027-6050
MSC (2010): Primary 57N10
Published electronically: June 30, 2014
MathSciNet review: 3256192
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that in any triangulation of a solid torus there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid torus. This theorem imposes restrictions on the possible Riemannian metrics on a solid torus. It also has applications in knot theory.

References [Enhancements On Off] (What's this?)

  • [1] Wolfgang Haken, Theorie der Normalflächen, Acta Math. 105 (1961), 245-375 (German). MR 0141106 (25 #4519a)
  • [2] Joel Hass and Jeffrey C. Lagarias, The number of Reidemeister moves needed for unknotting, J. Amer. Math. Soc. 14 (2001), no. 2, 399-428 (electronic). MR 1815217 (2001m:57012),
  • [3] William Jaco and Ulrich Oertel, An algorithm to decide if a $ 3$-manifold is a Haken manifold, Topology 23 (1984), no. 2, 195-209. MR 744850 (85j:57014),
  • [4] William Jaco, Hyam Rubinstein, and Stephan Tillmann, Minimal triangulations for an infinite family of lens spaces, J. Topol. 2 (2009), no. 1, 157-180. MR 2499441 (2010b:57016),
  • [5] Marc Lackenby, Word hyperbolic Dehn surgery, Invent. Math. 140 (2000), no. 2, 243-282. MR 1756996 (2001m:57003),
  • [6] Marc Lackenby, The crossing number of composite knots, J. Topol. 2 (2009), no. 4, 747-768. MR 2574742 (2011e:57013),
  • [7] Marc Lackenby, The crossing number of satellite knots, arxiv:1106.3095.
  • [8] Sergei Matveev, Algorithmic topology and classification of 3-manifolds, Algorithms and Computation in Mathematics, vol. 9, Springer-Verlag, Berlin, 2003. MR 1997069 (2004i:57026)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57N10

Retrieve articles in all journals with MSC (2010): 57N10

Additional Information

Marc Lackenby
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, United Kingdom

Received by editor(s): February 2, 2012
Received by editor(s) in revised form: March 19, 2013
Published electronically: June 30, 2014
Article copyright: © Copyright 2014 by the author

American Mathematical Society