On the degeneration of tunnel numbers under a connected sum
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- by Tao Li and Ruifeng Qiu PDF
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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 $t(k\# k’)\le t(k)$. This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel numbers under a connected sum and meridionally primitive knots.References
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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
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 2793-2807
- MSC (2010): Primary 57M25, 57N10
- DOI: https://doi.org/10.1090/tran/6473
- MathSciNet review: 3449258