Algebraic & Geometric Topology

A lower bound on tunnel number degeneration

Trenton Schirmer

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We prove a theorem that bounds the Heegaard genus from below under special kinds of toroidal amalgamations of 3–manifolds. As a consequence, we conclude that t(K1 # K2) max{t(K1),t(K2)} for any pair of knots K1,K2 S3, where t(K) denotes the tunnel number of K.

Article information

Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1279-1308.

Received: 3 December 2012
Revised: 3 August 2015
Accepted: 7 August 2015
First available in Project Euclid: 28 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

tunnel number knots Heegaard splittings connected sum


Schirmer, Trenton. A lower bound on tunnel number degeneration. Algebr. Geom. Topol. 16 (2016), no. 3, 1279--1308. doi:10.2140/agt.2016.16.1279.

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  • M Boileau, H Zieschang, Heegaard genus of closed orientable Seifert $3$–manifolds, Invent. Math. 76 (1984) 455–468
  • F Bonahon, J-P Otal, Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup. 16 (1983) 451–466
  • T Kobayashi, A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum, J. Knot Theory Ramifications 3 (1994) 179–186
  • T Kobayashi, Y Rieck, Heegaard genus of the connected sum of $m$–small knots, Comm. Anal. Geom. 14 (2006) 1037–1077
  • T Kobayashi, Y Rieck, Knot exteriors with additive Heegaard genus and Morimoto's conjecture, Algebr. Geom. Topol. 8 (2008) 953–969
  • T Li, Rank and genus of $3$–manifolds, J. Amer. Math. Soc. 26 (2013) 777–829
  • T Li, R Qiu, On the degeneration of tunnel numbers under a connected sum, Trans. Amer. Math. Soc. 368 (2016) 2793–2807
  • Y Moriah, H Rubinstein, Heegaard structures of negatively curved $3$–manifolds, Comm. Anal. Geom. 5 (1997) 375–412
  • K Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995) 3527–3532
  • K Morimoto, M Sakuma, Y Yokota, Examples of tunnel number one knots which have the property “$1+1=3$”, Math. Proc. Cambridge Philos. Soc. 119 (1996) 113–118
  • K Morimoto, J Schultens, Tunnel numbers of small knots do not go down under connected sum, Proc. Amer. Math. Soc. 128 (2000) 269–278
  • J,M Nogueira, Tunnel number degeneration under the connected sum of prime knots, Topology Appl. 160 (2013) 1017–1044
  • F,H Norwood, Every two-generator knot is prime, Proc. Amer. Math. Soc. 86 (1982) 143–147
  • T Saito, M Scharlemann, J Schultens, Lecture notes on generalized Heegaard splittings, preprint (2005)
  • M Scharlemann, J Schultens, The tunnel number of the sum of $n$ knots is at least $n$, Topology 38 (1999) 265–270
  • M Scharlemann, J Schultens, Annuli in generalized Heegaard splittings and degeneration of tunnel number, Math. Ann. 317 (2000) 783–820
  • M Scharlemann, A Thompson, Thin position for $3$–manifolds, from: “Geometric topology”, (C Gordon, Y Moriah, B Wajnryb, editors), Contemp. Math. 164, Amer. Math. Soc. (1994) 231–238
  • J Schultens, Additivity of tunnel number for small knots, Comment. Math. Helv. 75 (2000) 353–367