Geometry & Topology

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

Robert E Gompf, Martin Scharlemann, and Abigail Thompson

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

Permanent link to this document
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

Keywords
Property R Slice-Ribbon Conjecture Andrews–Curtis moves

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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