#### Volume 16, issue 3 (2016)

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A lower bound on tunnel number degeneration

### Trenton Schirmer

Algebraic & Geometric Topology 16 (2016) 1279–1308
##### 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 $K$.

##### Keywords
tunnel number, knots, Heegaard splittings, connected sum
##### Mathematical Subject Classification 2010
Primary: 57M25, 57N10