Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures

TL;DR

This paper investigates potential counterexamples to the Generalized Property R Conjecture, focusing on fibered knots and links containing the square knot, and explores their implications for slice and ribbon knot conjectures.

Contribution

It characterizes all two-component links with the square knot that surger to a connected sum of two S^1 x S^2, and identifies a family of likely counterexamples to the conjecture.

Findings

Identified all two-component links with the square knot that surger to (S^1 x S^2) # (S^1 x S^2)

Exhibited a family of links that are probable counterexamples to the Generalized Property R Conjecture

Generated slice knots that are not known to be ribbon

Abstract

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S^1 x S^2) # (S^1 x S^2). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.