Knot theory of Morse-Bott critical loci

TL;DR

This paper provides an alternative proof that critical knots of Morse-Bott functions on the 3-sphere are graph knots, using induction on the number of index-1 critical knots, contributing to the understanding of knot types in Morse theory.

Contribution

It introduces a new inductive proof method for classifying critical knots of Morse-Bott functions on S^3 as graph knots, expanding the theoretical framework.

Findings

Critical knots of Morse-Bott functions on S^3 are graph knots.

The proof uses induction on index-1 critical knots.

Provides a new perspective on the topology of Morse-Bott critical loci.

Abstract

We give an alternative proof of that a critical knot of a Morse-Bott function $f: S^3 \rightarrow \mathbb{R}$ is a graph knot where the critical set of $f$ is a link in $S^3$. Our proof inducts on the number of index-1 critical knots of $f$.