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High distance tangles and tunnel number of knots

  • TOSHIO SAITO (a1)
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 = T 1T 2 such that tnl(Ti ) = ti (i = 1, 2) and that tnl(K) = tnl(T 1) + tnl(T 2) + 2n − 1, where tnl(⋅) is the tunnel number. This contains an affirmative answer to an unsolved problem asked by Morimoto.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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