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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

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(S1 × S2). 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.

Property R, Slice-Ribbon Conjecture, Andrews–Curtis moves
Received: 21 January 2010
Revised: 24 August 2010
Accepted: 29 September 2010
Published: 20 November 2010
Proposed: Rob Kirby
Seconded: Mike Freedman, Colin Rourke
Robert E Gompf
Mathematics Department
University of Texas at Austin
1 University Station C1200
Austin TX 78712-0257
Martin Scharlemann
Mathematics Department
University of California, Santa Barbara
Santa Barbara CA 93106
Abigail Thompson
Mathematics Department
University of California, Davis
Davis CA 95616