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On the degeneration of tunnel numbers under a connected sum


Authors: Tao Li and Ruifeng Qiu
Journal: Trans. Amer. Math. Soc. 368 (2016), 2793-2807
MSC (2010): Primary 57M25, 57N10
DOI: https://doi.org/10.1090/tran/6473
Published electronically: June 15, 2015
MathSciNet review: 3449258
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that, for any integer $ n\ge 3$, there is a prime knot $ k$ such that (1) $ k$ is not meridionally primitive, and (2) for every $ m$-bridge knot $ k'$ with $ m\leq n$, the tunnel numbers satisfy $ t(k\char93 k')\le t(k)$. This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel numbers under a connected sum and meridionally primitive knots.


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Additional Information

Tao Li
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
Email: taoli@bc.edu

Ruifeng Qiu
Affiliation: Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, People’s Republic of China
Email: rfqiu@math.ecnu.edu.cn

DOI: https://doi.org/10.1090/tran/6473
Received by editor(s): December 5, 2013
Received by editor(s) in revised form: April 10, 2014
Published electronically: June 15, 2015
Additional Notes: The first author was partially supported by NSF grants DMS-1005556 and DMS-1305613. The second author was partially supported by NSFC 11171108.
Article copyright: © Copyright 2015 American Mathematical Society

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