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

2010/11/20 by Robert E. Gompf, Martin Scharlemann, Abigail Thompson · 1 citation
Mathematics · Medicine · #Botulinum Toxin and Related Neurological Disorders #Combinatorics #Conjecture #Counterexample #Discrete mathematics #Fibered knot #Geometric and Algebraic Topology #Geometry #Knot (papermaking) #Knot invariant #Knot theory #Mathematics #Monodromy #Property (philosophy) #Pure mathematics #Quantum invariant #Ribbon

paper · pdf · doi:10.2140/gt.2010.14.2305

openalex publication_date 2010/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11

Abstract

If there are any [math] –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.\n¶ The simplest plausible counterexample to the Generalized Property R Conjecture could be a [math] –component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to [math] . 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.

Citations

• The topology of four-dimensional manifolds
• Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems
• Man and machine thinking about the smooth 4-dimensional Poincaré conjecture
• Reducing heegaard splittings
• More Cappell–Shaneson spheres are standard
• Cappell-Shaneson homotopy spheres are standard

Cited by

• Ribbon knot [wikipedia]