## Geometry & Topology

### 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 $#2(S1×S2)$. 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.

#### Article information

Source
Geom. Topol., Volume 14, Number 4 (2010), 2305-2347.

Dates
Revised: 24 August 2010
Accepted: 29 September 2010
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883521

Digital Object Identifier
doi:10.2140/gt.2010.14.2305

Mathematical Reviews number (MathSciNet)
MR2740649

Zentralblatt MATH identifier
1214.57008

#### Citation

Gompf, Robert E; Scharlemann, Martin; Thompson, Abigail. Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures. Geom. Topol. 14 (2010), no. 4, 2305--2347. doi:10.2140/gt.2010.14.2305. https://projecteuclid.org/euclid.gt/1513883521

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