## The Michigan Mathematical Journal

### 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 $C(N)=S^{k+2}\setminus N$. Denote by $\xi\in H^{1}(C(N))$ the class dual to $N$. The Morse–Novikov number of $C(N)$ is by definition the minimal possible number of critical points of a regular Morse map $C(N)\to S^{1}$ 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.

#### Article information

Source
Michigan Math. J., Volume 66, Issue 4 (2017), 813-830.

Dates
Revised: 2 May 2017
First available in Project Euclid: 24 October 2017

https://projecteuclid.org/euclid.mmj/1508810816

Digital Object Identifier
doi:10.1307/mmj/1508810816

Mathematical Reviews number (MathSciNet)
MR3720325

Zentralblatt MATH identifier
06822187

#### Citation

Endo, Hisaaki; Pajitnov, Andrei. Circle-Valued Morse Theory for Frame Spun Knots and Surface-Links. Michigan Math. J. 66 (2017), no. 4, 813--830. doi:10.1307/mmj/1508810816. https://projecteuclid.org/euclid.mmj/1508810816

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