On the Morse-Novikov number for 2-knots

Abstract

Let $K\subset S^4$ be a 2-knot, that is, a smoothly embedded 2-sphere in $S^4$. 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 $\le 2\mathcal{M}\mathcal{N}(K)$. This enables us to compute $\mathcal{M}\mathcal{N}(S(K))$ for every classical knot $K$ with tunnel number 1.

Latex, 14 pages