An overview of property 2R. (English) Zbl 1221.57012

Scharlemann, Martin

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].

Reviewer: Claus Ernst (Bowling Green)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)

Keywords:

Property \(R\); Property \(2R\)

Citations:

Zbl 0627.57012; Zbl 1214.57008