Circle-valued Morse theory for frame spun knots and surface-links

TL;DR

This paper explores the Morse-Novikov number for complements of certain knots and surface-links, providing formulas that connect these invariants for spun knots and applying them to surface-links in 4-spheres.

Contribution

It introduces a formula relating Morse-Novikov numbers of twist frame spun knots to their base knots, extending classical results and applying to surface-links in 4-spheres.

Findings

Derived a formula linking Morse-Novikov numbers of spun knots and their base knots.

Extended classical results on fibrations of spun knots.

Computed Morse-Novikov numbers for surface-links in 4-spheres.

Abstract

Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), 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 f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which 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.