On the Morse-Novikov number for 2-knots

TL;DR

This paper investigates the Morse-Novikov number for 2-knots, establishing a relationship with classical knots and providing computations for certain spun knots, thereby advancing understanding of knot complexity in 4-dimensional space.

Contribution

It proves an inequality relating the Morse-Novikov number of spun 2-knots to classical knots and computes this number for all classical knots with tunnel number 1.

Findings

Morse-Novikov number of spun knots is at most twice that of the original knot.

Computed Morse-Novikov numbers for all classical knots with tunnel number 1.

Established a method to relate 2-knot invariants to classical knot invariants.

Abstract

Let $K\subset S^4$ be a 2-knot, that is, a smoothly embedded 2-sphere in $S^4$. The Morse-Novikov number $\mathcal M\mathcal N(K)$ is the minimal possible number of critical points of a Morse map $S^4\setminus K\to S^1$ belonging to the canonical class in $H^1(S^4\setminus K)$. We prove that for a classical knot $K\subset S^3$ the Morse-Novikov number of the spun knot $S(K)$ is $\leq 2\mathcal M\mathcal N(K)$. This enables us to compute $\mathcal M\mathcal N(S(K))$ for every classical knot $K$ with tunnel number 1.