Tunnel numbers of small knots do not go down under connected sum

Abstract

Let $K_1$ and $K_2$ be two knots in $S^3$ and $t(K_1)$, $t(K_2)$ the tunnel numbers of them. In this paper, we show that if both $K_1$ and $K_2$ are small, then $t(K_1 \# K_2) \ge t(K_1) + t(K_2)$. Moreover we show that $t(K_1 \# K_2 \# \cdots \# K_n) \ge t(K_1) + t(K_2) + \cdots + t(K_n)$ for any small knots $K_1, K_2, \cdots , K_n$.