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Proposed property 2R counterexamples examined. (English) Zbl 1376.57012
The property $$R$$ conjecture states that if $$0$$-surgery on a knot $$K \subset S^3$$ yields $$S^1 \times S^2$$, then $$K$$ is the unknot. This was verified by D. Gabai in [J. Differ. Geom. 26, 461–478 (1987; Zbl 0627.57012)]. The property $$2R$$ conjecture states that if $$0$$-surgery on a 2-component link $$L$$ yields $$\#^2 S^1 \times S^2$$, then band sums on $$L$$ yield the $$2$$-component unlink.
R. E. Gompf, the present author and A. Thompson [Geom. Topol. 14, No. 4, 2305–2347 (2010; Zbl 1214.57008)] provided potential counterexamples to the property $$2R$$ conjecture, denoted $$L_n$$, for $$n \in \mathbb{N}$$. If their examples do satisfy property $$2R$$, then certain presentations of the trivial group, that are thought to be Andrews-Curtis nontrivial when $$n \geq 3$$, would in fact be related to the trivial group by Andrews-Curtis moves. This is in turn related to the question of whether one can restrict to manifolds without 3-handles when investigating potential counterexamples to the smooth 4-dimensional Schoenflies or Poincaré conjectures.
Moreover the links $$L_n$$ are slice links, but are not known to be ribbon. Band sums of one component of $$L_n$$ over the other yields slice knots that are potential counter examples to the slice-ribbon conjecture.
The paper under review revisits the Gompf-Scharlemann-Thompson links $$L_n$$, filling in details of diagrammatic computations that were only sketched in [loc. cit.], and fleshing out the relation to the Andrews-Curtis conjecture. In particular the trace $$W$$ of surgery on $$L_n$$ is a cobordism from $$S^3$$ to $$\#^2 S^1 \times S^2$$ with $$\pi_1(W) \cong \langle a,b \mid aba = bab, a^n = b^{n+1} \rangle \cong \{1\}$$. The only known way to trivialise this presentation involves introducing extra relations. In the link language, this corresponds to introducing and cancelling Hopf pairs (i.e. cancelling 2- and 3-handle pairs) as well as band sums.
For anyone wishing to come to grips with the subtleties involving these potentially very important links and their relationship with several central conjectures in smooth 4-manifold theory, this paper should be essential companion reading to [loc. cit.].
##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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