On the degeneration of tunnel numbers under connected sum

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 $k^{\prime }$ with $m\le n$, the tunnel numbers satisfy $t(k\#k^{\prime })\le t(k)$. This gives counterexamples to a conjecture of Morimoto and Moriah on tunnel number under connected sum and meridionally primitive knots.

17 pages, 3 figures

References in corpus (2)

• Heegaard splittings of knot exteriors
• Tunnel number degeneration under the connected sum of prime knots