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Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures. (English) Zbl 1214.57008
The Property R theorem states that the \(0\)-framed surgery on any non-trivial knot in \(S^3\) does not give \(S^1\times S^2\), where \(S^k\) denotes the standard \(k\)-dimensional sphere [D. Gabai, J. Differ. Geom. 26, 479–536 (1987; Zbl 0639.57008)]. Problem 1.82 in (the new version of) Kirby’s problem list [Problems in low-dimensional topology. (Edited by Rob Kirby). Kazez, William H. (ed.), Geometric topology. 1993 Georgia international topology conference, August 2–13, 1993, Athens, GA, USA. Providence, RI: American Mathematical Society. AMS/IP Stud. Adv. Math. 2 (pt.2), 35–473 (1997; Zbl 0888.57014)] conjectures that if surgery on an \(n\)-component link in \(S^3\) gives the connected sum of \(n\) copies of \(S^1\times S^2\) then the link would become the \(0\)-framed unlink after handle slides (The Generalized Property R Conjecture). The authors propose the following conjecture (Property \(n\)R Conjecture) for knots: any knot in \(S^3\) cannot be a component of an \(n\)-component counterexample to the Generalized Property R Conjecture. Therefore the Generalized Property R Conjecture states that the Property \(n\)R Conjecture is true for all \(n\geq1\).
In the paper under review the authors give potential counterexamples to the Property \(2\)R Conjecture.
The authors first prove that a counterexample to the Property \(2\)R Conjecture with smallest genus is not fibered by showing that any counterexample to the Generalized Property R Conjecture with a fibered component gives another counterexample with smaller genus. They also show that the monodromy of a fibered counterexample has strong restrictions. In particular it is shown that for the square knot \(Q\), the connected sum of the trefoil and its mirror image, if \(Q\cup V\) gives a counterexample to the Generalized Property \(R\) Conjecture then the restrictions become simple enough to enumerate all such \(V\).
Among these the authors study the link \(L_{n,1}\) that consists of \(Q\) and the connected sum of the \((n,n+1)\)-torus knot and its mirror image. By using four-dimensional techniques it is shown that if the presentation \(\langle x,y\mid yxy=xyx,x^{n+1}=y^n\rangle\) of the trivial group gives a counterexample to the Andrews–Curtis conjecture [J. J. Andrews and M. L. Curtis, Proc. Am. Math. Soc. 16, 192–195 (1965; Zbl 0131.38301)], then \(L_{n,1}\) is a counterexample to the Generalized Property R Conjecture.
These examples also give slice knots that are not known to be ribbon giving potential counterexamples to the slice-ribbon problem (Problem 1.33 in Kirby’s problem list [loc.cit.]). Here a knot in \(S^3=\partial B^4\) is called slice if it bounds a smooth disk in \(B^4\) and is called ribbon if one can choose such a disk so that it has no local maxima with respect to the radial function.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N70 Cobordism and concordance in topological manifolds
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57R65 Surgery and handlebodies
20F05 Generators, relations, and presentations of groups
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