Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T03:24:25.702Z Has data issue: false hasContentIssue false
  • English
  • Français

High distance tangles and tunnel number of knots

Published online by Cambridge University Press:  14 August 2017

TOSHIO SAITO
TOSHIO SAITO*
Affiliation:
Department of Mathematics, Joetsu University of Education, 1 Yamayashiki, Joetsu 943-8512Japan. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for any integers ti ⩾ 0 (i = 1, 2) and n ⩾ 2, there is a knot K in the 3-sphere with an n-tangle decomposition K = T1∪T2 such that tnl(Ti) = ti (i = 1, 2) and that tnl(K) = tnl(T1) + tnl(T2) + 2n − 1, where tnl(⋅) is the tunnel number. This contains an affirmative answer to an unsolved problem asked by Morimoto.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 541 - 548
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Hartshorn, K. Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math. 204 (2002), no. 1, 61–75.Google Scholar
[2] Hempel, J. 3-manifolds as viewed from the curve complex. Topology 40 (2001), no. 3, 631–657.Google Scholar
[3] Jaco, W. Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43 (American Mathematical Society, 1980).Google Scholar
[4] Kobayashi, T. and Qiu, R.. The amalgamation of high distance Heegaard splittings is always efficient. Math. Ann. 341 (2008), 707–715.Google Scholar
[5] Minsky, Y. N. and Moriah, Y. and Schleimer, S.. High distance knots. Algebr. Geom. Topol. 7 (2007), 1471–1483.Google Scholar
[6] Morimoto, K. On the degeneration ratio of tunnel numbers and free tangle decompositions of knots. Geom. Topol. Monogr. 12 (2007), 265–275.Google Scholar
[7] Nogueira, J. M. Tunnel number degeneration under the connected sum of prime knots. Topology Appl. 160 (2013), no. 9, 1017–1044.Google Scholar
[8] Ochiai, M. On Haken's theorem and its extension. Osaka J. Math. 20 (1983), no. 2, 461–468.Google Scholar
[9] Saito, T. Tunnel number of tangles and knots. J. Math. Soc. Japan 66 (2014), no. 4, 1303–1313.Google Scholar
[10] Schultens, J. Additivity of tunnel number for small knots. Comment. Math. Helv. 75 (2000), no. 3, 353–367.Google Scholar
[11] Tomova, M. Thin position for knots in a 3-manifold. J. London Math. Soc. (2) 80 (2009), no. 1, 85–98.Google Scholar