Circle-Valued Morse Theory for Frame Spun Knots and Surface-Links

Abstract

Let $N^k\subset S^{k+2}$ be a closed oriented submanifold. Denote its complement by $C\left(N\right)=S^{k+2}\setminus N$. Denote by $\xi \in H^1\left(C\right(N\left)\right)$ the class dual to $N$. The Morse–Novikov number of $C\left(N\right)$ is by definition the minimal possible number of critical points of a regular Morse map $C\left(N\right)\to S^1$ belonging to $\xi$. In the first part of this paper, we study the case where $N$ is the twist frame spun knot associated with an $m$-knot $K$. We obtain a formula that relates the Morse–Novikov numbers of $N$ and $K$ and generalizes the classical results of D. Roseman and E. C. Zeeman about fibrations of spun knots. In the second part, we apply the obtained results to the computation of Morse–Novikov numbers of surface-links in 4-sphere.