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Examples of tunnel number one knots which have the property ‘1 + 1 = 3’

Published online by Cambridge University Press:  24 October 2008

Kanji Morimoto ,
Makoto Sakuma  and
Yoshiyuki Yokota
Kanji Morimoto
Affiliation:
Department of Mathematics, Takushoku University, Tatemachi, Hachioji, Tokyo 193, Japan
Makoto Sakuma
Affiliation:
Department of Mathematics, Faculty of Science. Osaka University, Machikaneyama-cho 1-16, Toyonaka, Osaka 560, Japan
Yoshiyuki Yokota
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka 812, Japan
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Extract

Let K be a knot in the 3-sphere S3, N(K) the regular neighbourhood of K and E(K) = cl(S3−N(K)) the exterior of K. The tunnel number t(K) is the minimum number of mutually disjoint arcs properly embedded in E(K) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K. In particular, we call it an unknotting tunnel if the system consists of a single arc.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 1 , January 1996 , pp. 113 - 118
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Boileau, M.. Lustig, M. and Moriah, Y.. Links with super-additive tunnel number (preprint).Google Scholar
[2]Boileau, M.. Rost, M. and Zieschang, H.. On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces. Math. Ann. 279 (1988), 553–581.CrossRefGoogle Scholar
[3]Doll, H.. A generalized bridge number for links in 3-manifolds. Math. Ann. 294 (1992), 701–717.CrossRefGoogle Scholar
[4]Haken, V.. Some reaultn on surface in 3-manifolds, Studies in Modern Topology, Math. Assoc. Amer. (Prentice-Hall. 1968).Google Scholar
[5]Jones, A. C.. Composite two-generator links have a Hopf link summand (preprint).Google Scholar
[6]Jokes, V. F. R.. Hecke algebra representations of braid groups and link polynomials. Ann. Math. 126 (1987), 335–388.Google Scholar
[7]Kauffman, L. H.. State models and the Jones polynomials. Topology 26 (1987), 395–407.CrossRefGoogle Scholar
[8]Kobayashi, T.. Arbitrarily highly degenerations of tunnel numbers of knots, to appear in Knot Ramifications.Google Scholar
[9]Kohno, T.. Tunnel number of knots and Jones-Whitten invariants (preprint).Google Scholar
[10]Montesinos, J. M.. Surgery on links and double branched covers of S3. Ann. Math. Studies 84 (1975), 227–259.Google Scholar
[11]Moriah, Y. and Rubinstein, H.. Heegaard structures of negatively curved 3-manifoIds (preprint).Google Scholar
[12]Morimoto, K.. On the additivity of tunnel number of knots. Topology Appl. 53 (1993), 37–66.CrossRefGoogle Scholar
[13]Morimoto, K.. There are knots whose tunnel numbers go down under connected sum, to appear in Proc. A M.S.Google Scholar
[14]Morimoto, K. and Sakuma, M.. On unknotting tunnels for knots. Math. Ann. 289 (1991), 143–167.CrossRefGoogle Scholar
[15]Norwood, F. H.. Every two generator knot is prime. Proc. A.M.S. 86 (1982), 143–147.CrossRefGoogle Scholar
[16]Scharlemann, M.. Tunnel number one knots satisfy the Poenaru conjecture. Topology Appl. 18 (1984), 235–258.CrossRefGoogle Scholar
[17]Yokota, Y.. Twisting formulae of the Jones polynomial. Math. Proc. Camb. Phil. Soc. 110 (1991), 473–482.CrossRefGoogle Scholar
[18]Yokota, Y.. On quantum SU(2) invariants and generalized bridge numbers of knots. Math. Proc. Camb. Phil. Soc. 117 (1995), 545–557.CrossRefGoogle Scholar