# On the degeneration ratio of tunnel numbers and free tangle decompositions of knots

@article{Morimoto2007OnTD, title={On the degeneration ratio of tunnel numbers and free tangle decompositions of knots}, author={K. Morimoto}, journal={arXiv: Geometric Topology}, year={2007} }

In this paper, we introduce a notion called n/k-free tangle and study the degeneration ratio of tunnel numbers of knots.

#### 4 Citations

Tunnel number degeneration under the connected sum of prime knots

- Mathematics
- 2013

Abstract We study 2-string free tangle decompositions of knots with tunnel number two. As an application, we construct infinitely many counter-examples to a conjecture in the literature stating that… Expand

High distance tangles and tunnel number of knots

- Mathematics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2017

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) +… Expand

Two varieties of tunnel number subadditivity

- Mathematics
- 2012

Knot theory and 3-manifold topology are closely intertwined, and few invariants stand so firmly in the intersection of these two subjects as the tunnel number of a knot, denoted t(K). We describe two… Expand

New examples of tunnel number subadditivity

- Mathematics
- 2011

Abstract If the tunnel number of a knot K is denoted t ( K ) , a pair of knots K 1 , K 2 is said to be subadditive if t ( K 1 ) + t ( K 2 ) > t ( K 1 # K 2 ) . Scharlemann and Schultens (2000) [11]… Expand

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- Mathematics
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- Mathematics
- 1995

In this paper, we show that there are infinitely many tunnel number two knots K such that the tunnel number of K#K' is equal to two again for any 2-bridge knot K'. INTRODUCTION Let K be a knot in the… Expand

Higashi-Nada Okamoto 8-9-1, Kobe 658-8501, Japan morimoto@konan-u.ac

- jp Received: 5 December