Michigan Mathematical Journal

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

Hisaaki Endo and Andrei Pajitnov

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Let NkSk+2 be a closed oriented submanifold. Denote its complement by C(N)=Sk+2N. Denote by ξH1(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)S1 belonging to ξ. 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

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

Received: 20 June 2016
Revised: 2 May 2017
First available in Project Euclid: 24 October 2017

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Zentralblatt MATH identifier

Primary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25} 57R35: Differentiable mappings 57R70: Critical points and critical submanifolds 57R45: Singularities of differentiable mappings


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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