Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Tunnel leveling, depth, and bridge numbers


Authors: Sangbum Cho and Darryl McCullough
Journal: Trans. Amer. Math. Soc. 363 (2011), 259-280
MSC (2010): Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-2010-05248-1
Published electronically: August 24, 2010
MathSciNet review: 2719681
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use the theory of tunnel number $ 1$ knots introduced in an earlier paper to strengthen the Tunnel Leveling Theorem of Goda, Scharlemann, and Thompson. This yields considerable information about bridge numbers of tunnel number $ 1$ knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant $ d$ of its tunnels. The growth of this value is on the order of $ (1+\sqrt{2})^d$, which improves known estimates of the rate of growth of bridge number as a function of the Hempel distance of the associated Heegaard splitting. We also find the maximum bridge number as a function of the number of cabling constructions needed to produce the tunnel, showing in particular that the maximum bridge number of a knot produced by $ n$ cabling constructions is the $ (n+2)^{nd}$ Fibonacci number. Finally, we examine the special case of the ``middle'' tunnels of torus knots.


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

  • 1. C. Adams and A. Reid, Unknotting tunnels in two-bridge knot and link complements, Comment. Math. Helv. 71 (1996), 617-627. MR 1420513 (98h:57009)
  • 2. E. Akbas, A presentation of the automorphisms of the $ 3$-sphere that preserve a genus two Heegaard splitting, Pacific J. Math. 236 (2008), 201-222. MR 2407105 (2009d:57029)
  • 3. M. Boileau, M. Rost, and H. Zieschang, On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces, Math. Ann. 279 (1988), 553-581. MR 922434 (89a:57013)
  • 4. S. Cho, Homeomorphisms of the $ 3$-sphere that preserve a Heegaard splitting of genus two, Proc. Amer. Math. Soc. 136 (2008), 1113-1123. MR 2361888 (2009c:57029)
  • 5. S. Cho and D. McCullough, The tree of knot tunnels, Geom. Topol. 13 (2009) 769-815. MR 2469530
  • 6. S. Cho and D. McCullough, Cabling sequences of tunnels of torus knots, Algebr. Geom. Topol. 9 (2009) 1-20. MR 2471129 (2009i:57011)
  • 7. S. Cho and D. McCullough, Constructing knot tunnels using giant steps, arXiv:0812.1382.
  • 8. S. Cho and D. McCullough, software available at www.math.ou.edu/ $ _{\widetilde{\phantom{n}}}$dmccullough/ .
  • 9. H. Goda, M. Ozawa, and M. Teragaito, On tangle decompositions of tunnel number one links, J. Knot Theory Ramifications 8 (1999), 299-320. MR 1691429 (2000d:57004)
  • 10. H. Goda, M. Scharlemann, and A. Thompson, Levelling an unknotting tunnel, Geom. Topol. 4 (2000), 243-275. MR 1778174 (2002h:57011)
  • 11. J. Johnson, Bridge number and the curve complex, Mathematics ArXiv math.GT/0603102.
  • 12. M. Kuhn, Tunnels of $ 2$-bridge links, J. Knot Theory Ramifications 5 (1996), 167-171. MR 1395777 (97g:57008)
  • 13. Y. Minsky, Y. Moriah, and S. Schleimer, High distance knots, Algebr. Geom. Topol. 7 (2007), 1471-1483. MR 2366166 (2008k:57016)
  • 14. Y. Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988), 465-481. MR 928492 (89d:57010)
  • 15. K. Morimoto and M. Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991), 143-167. MR 1087243 (92e:57015)
  • 16. M. Scharlemann, Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana (3) 10 (2004) 503-514. MR 2199366 (2007c:57020)
  • 17. M. Scharlemann and A. Thompson, Unknotting tunnels and Seifert surfaces, Proc. London Math. Soc. (3) 87 (2003), 523-544. MR 1990938 (2004e:57015)
  • 18. M. Scharlemann and M. Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006), 593-617. MR 2224466 (2007b:57040)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57M25

Retrieve articles in all journals with MSC (2010): 57M25


Additional Information

Sangbum Cho
Affiliation: Department of Mathematics, University of California at Riverside, Riverside, California 92521
Email: scho@math.ucr.edu

Darryl McCullough
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: dmccullough@math.ou.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05248-1
Received by editor(s): January 21, 2009
Published electronically: August 24, 2010
Additional Notes: The second author was supported in part by NSF grant DMS-0802424
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society