Equivalent characterizations of handle-ribbon knots

TL;DR

This paper characterizes handle-ribbon knots in homotopy 4-balls through the existence of specific derivatives and extends classical fibration theorems to handle-ribbon knots, advancing understanding of their structure.

Contribution

It establishes equivalent conditions for handle-ribbon knots involving R-link derivatives and extends Casson-Gordon's fibration theorem to non-fibered handle-ribbon knots.

Findings

Handle-ribbon knots admit R-link derivatives with specific properties.

A handle-ribbon disk exists if the zero-surgery manifold admits a suitable singular fibration.

Generalization of Casson-Gordon theorem to non-fibered handle-ribbon knots.

Abstract

The stable Kauffman conjecture posits that a knot in $S^3$ is slice if and only if it admits a slice derivative. We prove a related statement: A knot is handle-ribbon (also called strongly homotopy-ribbon) in a homotopy 4-ball $B$ if and only if it admits an R-link derivative; i.e. an $n$-component derivative $L$ with the property that zero-framed surgery on $L$ yields $\#^n(S^1\times S^2)$. We also show that $K$ bounds a handle-ribbon disk $D \subset B$ if and only if the 3-manifold obtained by zero-surgery on $K$ admits a singular fibration that extends over handlebodies in $B \setminus D$, generalizing a classical theorem of Casson and Gordon to the non-fibered case for handle-ribbon knots.