On the degeneration of tunnel numbers under a connected sum

Li, Tao; Qiu, Ruifeng

doi:10.1090/tran/6473

On the degeneration of tunnel numbers under a connected sum
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by Tao Li and Ruifeng Qiu PDF

Trans. Amer. Math. Soc. 368 (2016), 2793-2807 Request permission

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.

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