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

Gompf, Robert E¹ ; Scharlemann, Martin² ; Thompson, Abigail³

¹ Mathematics Department, University of Texas at Austin, 1 University Station C1200, Austin TX 78712-0257, USA
² Mathematics Department, University of California, Santa Barbara, Santa Barbara CA 93106, USA
³ Mathematics Department, University of California, Davis, Davis CA 95616, USA

Geometry & topology, Tome 14 (2010) no. 4, pp. 2305-2347.

Résumé

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.

Détail

DOI : 10.2140/gt.2010.14.2305

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

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