On the tunnel number and the Morse–Novikov number of knots

Abstract

Let $L$ be a link in $S^3$; denote by $\mathcal{ℳ}\mathcal{N}\left(L\right)$ the Morse–Novikov number of $L$ and by $t\left(L\right)$ the tunnel number of $L$. We prove that $\mathcal{ℳ}\mathcal{N}\left(L\right)\le 2t\left(L\right)$ and deduce several corollaries.