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The spectrum of the growth rate of the tunnel number is infinite


Authors: Kenneth L. Baker, Tsuyoshi Kobayashi and Yo’av Rieck
Journal: Proc. Amer. Math. Soc. 144 (2016), 3609-3618
MSC (2010): Primary 57M99, 57M25
DOI: https://doi.org/10.1090/proc/12957
Published electronically: February 2, 2016
MathSciNet review: 3503730
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Abstract: For any $ \epsilon > 0$ we construct a hyperbolic knot $ K \subset S^{3}$ for which $ 1 - \epsilon < \mathrm {gr}_t(K) < 1$. This shows that the spectrum of the growth rate of the tunnel number is infinite.


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Additional Information

Kenneth L. Baker
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33146
Email: k.baker@math.miami.edu

Tsuyoshi Kobayashi
Affiliation: Department of Mathematics, Nara Women’s University, Kitauoya Nishimachi, Nara 630-8506, Japan
Email: tsuyoshi@cc.nara-wu.ac.jp

Yo’av Rieck
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: yoav@uark.edu

DOI: https://doi.org/10.1090/proc/12957
Keywords: 3-manifold, knots, Heegaard splittings, tunnel number
Received by editor(s): July 13, 2015
Received by editor(s) in revised form: September 10, 2015
Published electronically: February 2, 2016
Additional Notes: The first and third authors would like to thank Nara Women’s University for their hospitality during the development of this article
The second author was supported by Grant-in-Aid for scientific research, JSPS grant number 00186751.
This work was partially supported by grants from the Simons Foundation (#209184 to the first author and #283495 to the third author)
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2016 American Mathematical Society