A lower bound on tunnel number degeneration

We prove a theorem which bounds Heegaard genus from below under special kinds of toroidal amalgamations of $3$-manifolds. As a consequence, we conclude $t(K_1\mathrm{\#}{K}_2)\ge \max \{t(K_1),t(K_2)\}$ for any pair of knots $K_1,K_2\subset S^3$,where$t(K)$denotesthetunnelnumberof$K$.