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An overview of property 2R. (English) Zbl 1221.57012
Banagl, Markus (ed.) et al., The mathematics of knots. Theory and application. Berlin: Springer (ISBN 978-3-642-15636-6/hbk; 978-3-642-15637-3/ebook). Contributions in Mathematical and Computational Sciences 1, 317-325 (2011).
By a Theorem of D. Gabai’s [J. Differ. Geom. 26, 461–478 (1987; Zbl 0627.57012)] all knots have property $$R$$, that is, if $$0$$-framed surgery on a knot $$K\in S^3$$ yields $$S^1\times S^2$$ then $$K$$ is the unknot. This Theorem leads to a Conjecture called the Generalized Property $$R$$ Conjecture. Suppose $$L$$ is an integrally framed link of $$n\geq 1$$ components in $$S^3$$, and surgery on $$L$$ via the specified framing yields $$\#_n(S^1\times S^2)$$. Then there is a sequence of handle slides on $$L$$ that converts $$L$$ into a $$0$$-framed unlink. Using this we define: A knot in $$S^3$$ has Property $$nR$$ if it does not appear among the components of any $$n$$-component counterexamples to the Generalized Property $$R$$ Conjecture. The author conjectures that all knots have Property $$nR$$. In this article the author gives an overview of the special case for $$n=2$$ called the property $$2R$$. The overview is based on the article of R. Gompf, A. Thompson and the author [Geom. Topol. 14, No. 4, 2305–2347 (2010; Zbl 1214.57008)].
For the entire collection see [Zbl 1205.57002].
##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)
##### Keywords:
Property $$R$$; Property $$2R$$
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