Communications in Analysis and Geometry

Volume 31 (2023)

Number 9

Equivalent characterizations of handle-ribbon knots

Pages: 2157 – 2193

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n9.a1

Authors

Maggie Miller (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Alexander Zupan (Department of Mathematics, University of Nebraska, Lincoln, Neb., U.S.A.)

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 homotopyribbon) 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 $B \setminus D$. This can be viewed as a version of a classical theorem of Casson and Gordon for homotopy-ribbon fibered knots, here extended to handle-ribbon knots that need not be fibered.

Received 15 September 2021

Accepted 12 October 2021

Published 12 August 2024