On the Morse-Novikov number for 2-Knots

Abstract

Let $K\subset S^4$ be a 2-knot. The Morse-Novikov number ${\mathcal M}{\mathcal N}(K)$ is the minimal possible number of critical points of a Morse map $S^4\setminus K\to S^1$ belonging to the canonical class in $H^1(S^4\setminus K)$. We prove that for a classical knot $K\subset S^3$ the Morse-Novikov number of the spun knot $S(K)$ is $\leq 2{\mathcal M}{\mathcal N}(K)$. This enables us to compute ${\mathcal M}{\mathcal N}(S(K))$ for every classical knot $K$ with tunnel number 1.