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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]
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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 

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References

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