Proposed Property 2R counterexamples classified

TL;DR

This paper clarifies the classification of certain two-component links containing the square knot, specifically analyzing the family $L_n$, and explores their implications for the Generalized Property R Conjecture.

Contribution

It provides detailed insights into the construction and classification of the $L_n$ links, addressing their role in the context of Property R conjectures.

Findings

The family $L_n$ links are classified within the existing framework.

The paper offers new perspectives on the construction of $L_n$ links.

Implications for the Generalized Property R Conjecture are discussed.

Abstract

Earlier work with Robert Gompf and Abigail Thompson classified, via a natural slope indexed by the rationals, all two-component links which contain the square knot and from which $(S^1 \times S^2) \# (S^1 \times S^2)$ can be obtained by surgery. It was argued that a certain family $L_n$ of such links probably contradict the Generalized Property R Conjecture. Left unresolved was how the family $L_n$ fits into the classification scheme. This question is resolved here, in part by giving varied perspectives and more detail on the construction of the $L_n$.