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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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