A lower bound on tunnel number degeneration

Abstract

We prove a theorem that bounds the Heegaard genus from below under special kinds of toroidal amalgamations of $3$–manifolds. As a consequence, we conclude that $t\left(K_1\#K_2\right)\ge max\left\{t\left(K_1\right),t\left(K_2\right)\right\}$ for any pair of knots $K_1,K_2\subset S^3$, where $t\left(K\right)$ denotes the tunnel number of $K$.