The Michigan Mathematical Journal

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

Hisaaki Endo and Andrei Pajitnov

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1508810816

Digital Object Identifier
doi:10.1307/mmj/1508810816

Mathematical Reviews number (MathSciNet)
MR3720325

Zentralblatt MATH identifier
06822187

Subjects
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

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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