Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Two-generator cable knots are tunnel one


Author: Steven A. Bleiler
Journal: Proc. Amer. Math. Soc. 122 (1994), 1285-1287
MSC: Primary 57M25; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9939-1994-1242075-3
MathSciNet review: 1242075
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A two-generator cable knot exterior is a genus two handlebody with a single two-handle attached.


References [Enhancements On Off] (What's this?)

  • [B] R. Bieri, Homological dimension of discrete groups, Queen Mary College Math. Notes, Queen Mary College, London, 1976. MR 0466344 (57:6224)
  • [C] B. Clark, The Heegaard genus of manifolds obtained by surgery on knots and links, Internat. J. Math. Math. Sci. 3 (1980), 583-589. MR 582900 (81j:57004)
  • [CF] R. Crowell and R. Fox, Introduction to knot theory, Graduate Texts in Math., vol. 57, Springer-Verlag, New York, 1963. MR 0146828 (26:4348)
  • [CGLS] M. Culler, C. Gordon, J. Luecke, and P. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), 237-300. MR 881270 (88a:57026)
  • [E] M. Eudave-Munoz, On non-simple 3-manifolds and 2-handle addition, preprint. MR 1256216 (95e:57029)
  • [G] C. Gordon, Dehn surgery on satellite knots, Trans. Amer. Math. Soc. 275 (1983), 687-708. MR 682725 (84d:57003)
  • [K] R. Kirby, Problems in low dimensional topology, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, RI, 1978, pp. 273-312. MR 520548 (80g:57002)
  • [M] Y. Moriah, A note on satellites and tunnel number, Kobe J. Math. 8 (1991), 73-79. MR 1134706 (92m:57014)
  • [MKS] W. Magus, A. Karass, and D. Solitar, Combinatorial group theory, Interscience, New York, London, and Sydney, 1966.
  • [MS] K. Marimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), 143-167. MR 1087243 (92e:57015)
  • [N1] F. Norwood, Every one relator knot is prime, preprint, 1979.
  • [N2] -, Every two-generator knot is prime, Proc. Amer. Math. Soc. 86 (1982), 143-147. MR 663884 (83k:57005)
  • [S] M. Scharlemann, Tunnel number one knots satisfy the Poenaru conjecture, Topology Appl. 18 (1984), 235-258. MR 769294 (86e:57009)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57M25, 57M05

Retrieve articles in all journals with MSC: 57M25, 57M05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1242075-3
Keywords: Two-generator knot, tunnel number, tunnel number one knot
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society