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Journal of the London Mathematical Society Advance Access published online on May 14, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp009
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© 2009 London Mathematical Society

Thin position for knots in a 3-manifold

Maggy Tomova

Mathematics Department
The University of Iowa
IA 52440
USA

We extend the notion of a thin position of a link in a 3-manifold with respect to a generalized Heegaard splitting introduced in Hayashi and Shimokawa (Pacific J. Math. 197 (2001) 301–324), to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that, if P is a c-weakly reducible bridge surface for a link K contained in a closed, orientable, irreducible 3-manifold M, then one of the following is satisfied:

  • P is stabilized;
  • P is meridionally stabilized;
  • P is perturbed;
  • a component of K is removable;
  • M contains an essential meridional surface.

2000 Mathematics Subject Classification 57M25, 57M27, 57M50.

This work was partially supported by an NSF grant.

Received April 17, 2007; revised October 1, 2008;
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