Volume 14, issue 4 (2010)

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

Robert E Gompf, Martin Scharlemann and Abigail Thompson

Geometry & Topology 14 (2010) 2305–2347
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 ${#}_{2}\left({S}^{1}×{S}^{2}\right)$. 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.

Keywords
Property R, Slice-Ribbon Conjecture, Andrews–Curtis moves