A lower bound on tunnel number degeneration

Abstract

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