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

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 ${\#}_2\left(S^1\times S^2\right)$. 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.