On the tunnel number and the Morse-Novikov number of knots (Q969656)

For a link in the 3-sphere, the minimal number of critical points over all possible circle-valued Morse maps of its exterior which behave well near the boundary is called the \textit{Morse-Novikov number} of the link. In the paper under review, the author gives an upper bound on the Morse-Novikov number in terms of the tunnel number: It is proved that the Morse-Novikov number is less than or equal to twice the tunnel number of the link. The proof uses a perturbation argument to construct a circle-valued Morse map from a given \(\mathbb{R}\)-valued Morse function of a link exterior. In fact, this technique would be applicable to any compact 3-manifold with toral boundary. There are several corollaries. In particular, it follows that the Morse-Novikov number of every tunnel number one knot is at most two, which gives an alternative proof of the fact that every two-bridge knot has Morse-Novikov number at most two, which was announced by M. Hirasawa.