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ISSN 1088-6850(e) ISSN 0002-9947(p)

Tunnel leveling, depth, and bridge numbers

Author(s): Sangbum Cho; Darryl McCullough
Journal: Trans. Amer. Math. Soc. 363 (2011), 259-280.
MSC (2010): Primary 57M25
Posted: August 24, 2010
MathSciNet review: 2719681
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Abstract | References | Similar articles | Additional information

Abstract: We use the theory of tunnel number 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 knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant 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 cabling constructions is the $(n+2)^{nd}$ Fibonacci number. Finally, we examine the special case of the ``middle'' tunnels of torus knots.

References:

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

-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

-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/ 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

-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)

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