Some experimental results on stable equivalence of GST Links for the Generalized Property R Conjecture

TL;DR

This paper implements an algorithm to explicitly construct Gompf-Scharlemann-Thompson and Meier-Zupan's R-links, verifying their stable handleslide triviality and equivalence, contributing to the study of the generalized property R conjecture.

Contribution

It introduces an algorithm for constructing these R-links and verifies their stable handleslide triviality and equivalence, advancing understanding of the conjecture.

Findings

Some links are stably handleslide trivial.

Many links are stably handleslide equivalent.

Results are independently confirmed in other recent work.

Abstract

Gompf-Scharlemann-Thompson and Meier-Zupan constructed an infinite family of R-links that are potential counterexamples of the generalized property R conjecture. Their works also show that whether these links are stably handleslide trivial is an interesting open problem related to the Slice-Ribbon conjecture. In this work, we implement an algorithm to construct all these links explicitly, the details of this algorithm will the content of another paper. With such an algorithm, the stable handleslide triviality of some of these links is verified. Moreover, many links are shown to be stably handleslide equivalent. Some of the results are obtained independently in \cite{Knots in the fiber}