On the degeneration of tunnel numbers under connected sum

TL;DR

This paper constructs specific prime knots demonstrating that tunnel numbers can degenerate under connected sum, providing counterexamples to a longstanding conjecture in knot theory.

Contribution

It introduces prime knots that are not meridionally primitive yet exhibit tunnel number degeneration under connected sum with certain knots.

Findings

Counterexamples to Morimoto and Moriah's conjecture

Existence of prime knots with tunnel number degeneration

Tunnel number behavior for m-bridge knots

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 number under connected sum and meridionally primitive knots.