Identifying tunnel number one knots

Kanji MORIMOTO; Makoto SAKUMA; Yoshiyuki YOKOTA

doi:10.2969/jmsj/04840667

Kanji MORIMOTO, Makoto SAKUMA, Yoshiyuki YOKOTA

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JOURNAL FREE ACCESS

1996 Volume 48 Issue 4 Pages 667-688

DOI https://doi.org/10.2969/jmsj/04840667

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Published: 1996 Received: July 29, 1994 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : AHW) C. Adams, M. Hildebrand and J. Weeks, Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc., 326 (1991), 1-56.
BGM) J.S. Birman, F. González-Acuna and H.M. Montesinos, Heegaard splittings of prime 3-manifolds are not unique, Michigan Math. J., 23 (1976), 97-103.
BH) J.S. Birman and H.M. Hilden, Heegaard splittings and branched coverings of S³, Trans. Amer. Math. Soc., 213 (1975), 315-352.
BM) S.A. Bleiler and Y. Moriah, Heegaard splittings and branched coverings of B³, Math. Ann., 281 (1988), 531-543.
BS) F. Bonahon and L. Siebenmann, Geometric splittings of links, and Conway's algebraic knots, preprint.
BoZm) M. Boileau and B. Zimmermann, Symmetries of nonelliptic Montesinos links, Math. Ann., 277 (1987), 563-584.
BoZe) M. Boileau and H. Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds, Invent. Math., 76 (1984), 455-468.
BuZ) G. Burde and H. Zieschang, Knots, De Gruyter Studies in Mathematics 5, de Gruyter, Berlin, New York, 1985.
D) H. Doll, A generalized bridge number for links in 3-manifolds, Math. Ann., 294 (1992), 701-717.
G) S. Garoufalidis, Application of TQFT invariants in low dimensional topology, preprint.
GS) F. González-Acuna and H. Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc., 99 (1986), 89-102.
HW) S. Henry and J. Weeks, Symmetry groups of hyperbolic knots and links, J. Knot Theory and its Ramification, 1 (1992), 185-201.
Ka) L.H. Kauffman, State models and the Jones polynomials, Topology, 26 (1987), 395-497.
KI) E. Klimenko, A class of 2-generator subgroups of PSL(2, C), Sibirsk. Mat. Zh., 30 (1989), 74-76.
Kd) K. Kodama, personal communication.
KS) K. Kodama and M. Sakuma, Symmetry groups of prime knots up to 10 crossings, Knots 90, (ed. A. Kawauchi), pp. 323-340.
Kh) T. Kohno, Tunnel number of knots and Jones-Witten invariants, Braid group, knot theory and statistical mechanics II, (ed. C.N. Yang and M.L. Ge), pp. 275-293.
L1) W.B.R. Lickorish, Skeins and handlebodies, Pacific J. Math., 159 (1993), 337-349.
L2) W.B.R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory and its Ramification, 2 (1993), 171-194.
Ma) J.P. Matelski, The classification of discrete 2-generator subgroups of PSL(2, R), Israel J. Math., 42 (1982), 309-317.
Mo) Y. Moriah, Heegaard splittings of Seifert fibred spaces, Invent. Math., 91 (1988), 465-481.
MS) K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann., 289 (1991), 143-167.
MSY) K. Morimoto, M. Sakuma and Y. Yokota, Examples of tunnel number one knots which have the property '1+1=3', Math. Proc. Cambridge Philos. Soc., 119 (1996), 113-118.
Sa) M. Sakuma, On strongly invertible knots, Algebraic and Topological Theories, Proc. Miyata memorial Conf., Kinokuniya Co. Ltd., 1985.
Sc) M. Scharlemann, Tunnel number one knots satisfy the Poenaru conjecture, Topology Appl., 18 (1984), 235-258.
T) J.L. Tollefson, Involutions of sufficiently large 3-manifolds, Topology, 20 (1981), 323-352.
V) O. Viro, Linkings, two sheeted branched coverings and braids, Math. USSR-Sb., 16 (1972), 223-235.
Wd) F. Waldhausen, Über Involutionen der 3-sphäre, Topology, 8 (1969), 81-91.
Wk) K. Walker, On Witten's 3-manifold invariants, preprint.
Ya) S. Yamada, A topological invariant of spatial regular graphs, Knots 90, Walter de Gruyter, 1992, pp. 447-454.
Yo1) Y. Yokota, On quantum SU(2) invariants and generalized bridge numbers of knots, Math. Proc. Cambridge Philos. Soc., 117 (1995), 545-557.
Yo2) Y. Yokota, Topological invariants of graphs in 3-space, Topology, 35 (1996), 77-87.
Z) H. Zieschang, Classification of Montesinos knots, Topology Proc. Leningrad 1982 Lecture Notes in Math., 1060, Springer-Verlag, 1984, pp. 378-389.

Right : [AHW] C. Adams, M. Hildebrand and J. Weeks, Hyperbolic invariants of knots and links, Trans. Amer. Math. Soc., 326 (1991), 1-56.
[BGM] J. S. Birman, F. González-Acuna and H. M. Montesinos, Heegaard splittings of prime 3-manifolds are not unique, Michigan Math. J., 23 (1976), 97-103.
[BH] J. S. Birman and H. M. Hilden, Heegaard splittings and branched coverings of S³, Trans. Amer. Math. Soc., 213 (1975), 315-352.
[BM] S. A. Bleiler and Y. Moriah, Heegaard splittings and branched coverings of B³, Math. Ann., 281 (1988), 531-543.
[BS] F. Bonahon and L. Siebenmann, Geometric splittings of links, and Conway's algebraic knots, preprint.
[BoZm] M. Boileau and B. Zimmermann, Symmetries of nonelliptic Montesinos links, Math. Ann., 277 (1987), 563-584.
[BoZe] M. Boileau and H. Zieschang, Heegaard genus of closed orientable Seifert 3-manifolds, Invent. Math., 76 (1984), 455-468.
[BuZ] G. Burde and H. Zieschang, Knots, De Gruyter Studies in Mathematics 5, de Gruyter, Berlin, New York, 1985.
[D] H. Doll, A generalized bridge number for links in 3-manifolds, Math. Ann., 294 (1992), 701-717.
[G] S. Garoufalidis, Application of TQFT invariants in low dimensional topology, preprint.
[GS] F. González-Acuna and H. Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc., 99 (1986), 89-102.
[HW] S. Henry and J. Weeks, Symmetry groups of hyperbolic knots and links, J. Knot Theory and its Ramification, 1 (1992), 185-201.
[Ka] L. H. Kauffman, State models and the Jones polynomials, Topology, 26 (1987), 395-497.
[Kl] E. Klimenko, A class of 2-generator subgroups of PSL (2, C), Sibirsk. Mat. Zh., 30 (1989), 74-76.
[Kd] K. Kodama, personal communication.
[KS] K. Kodama and M. Sakuma, Symmetry groups of prime knots up to 10 crossings, Knots 90, (ed. A. Kawauchi), pp. 323-340.
[Kh] T. Kohno, Tunnel number of knots and Jones-Witten invariants, Braid group, knot theory and statistical mechanics II, (ed. C. N. Yang and M. L. Ge), pp. 275-293.
[L1] W. B. R. Lickorish, Skeins and handlebodies, Pacific J. Math., 159 (1993), 337-349.
[L2] W. B. R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory and its Ramification, 2 (1993), 171-194.
[Ma] J. P. Matelski, The classification of discrete 2-generator subgroups of PSL (2, R), Israel J. Math., 42 (1982), 309-317.
[Mo] Y. Moriah, Heegaard splittings of Seifert fibred spaces, Invent. Math., 91 (1988), 465-481.
[MS] K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann., 289 (1991), 143-167.
[Sa] M. Sakuma, On strongly invertible knots, Algebraic and Topological Theories, Proc. Miyata memorial Conf., Kinokuniya Co. Ltd., 1985.
[Sc] M. Scharlemann, Tunnel number one knots satisfy the Poenaru conjecture, Topology Appl., 18 (1984), 235-258.
[T] J. L. Tollefson, Involutions of sufficiently large 3-manifolds, Topology, 20 (1981), 323-352.
[V] O. Viro, Linkings, two sheeted branched coverings and braids, Math. USSR-Sb., 16 (1972), 223-235.
[Wd] F. Waldhausen, Über Involutionen der 3-sphäre, Topology, 8 (1969), 81-91.
[Wk] K. Walker, On Witten's 3-manifold invariants, preprint.
[Ya] S. Yamada, A topological invariant of spatial regular graphs, Knots 90, Walter de Gruyter, 1992, pp. 447-454.
[Yo1] Y. Yokota, On quantum SU (2) invariants and generalized bridge numbers of knots, Math. Proc. Cambridge Philos. Soc., 117 (1995), 545-557.
[Yo2] Y. Yokota, Topological invariants of graphs in 3-space, Topology, 35 (1996), 77-87.
[Z] H. Zieschang, Classification of Montesinos knots, Topology Proc. Leningrad 1982 Lecture Notes in Math., 1060, Springer-Verlag, 1984, pp. 378-389.

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