Mathematische Annalen

, Volume 317, Issue 4, pp 783–820

Annuli in generalized Heegaard splittings and degeneration of tunnel number


  • Martin Scharlemann
    • Mathematics Department, University of California, Santa Barbara, CA 93106, USA (e-mail:
  • Jennifer Schultens
    • Department of Mathematics & CS, Emory University, Atlanta, GA 30322, USA (e-mail:
Original article

DOI: 10.1007/PL00004423

Cite this article as:
Scharlemann, M. & Schultens, J. Math Ann (2000) 317: 783. doi:10.1007/PL00004423

Abstract. We analyze how a family of essential annuli in a compact 3-manifold will induce, from a strongly irreducible generalized Heegaard splitting of the ambient manifold, generalized Heegaard splittings of the complementary components. There are specific applications to the subadditivity of tunnel number of knots, improving somewhat bounds of Kowng [Kw]. For example, in the absence of 2-bridge summands, the tunnel number of the sum of n knots is no less than \(\frac{2}{5}\) the sum of the tunnel numbers.

Mathematics Subject Classification (1991):57M25, 57M50, 57N10.
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© Springer-Verlag Berlin Heidelberg 2000