PROPOSED PROPERTY 2R COUNTEREXAMPLES CLASSIFIED

Abstract. [GST] classified, via a natural slope indexed by Q, all two-component links which contain the square knot and from which (S1 × S2)#(S1 × S2) can be obtained by surgery. It was argued there that each of a certain family Ln of such links probably con-tradicts the Generalized Property R Conjecture. Left unresolved was how the family Ln fits into the classification scheme. This question is resolved here, in part by giving varied perspectives and more detail on the construction of the Ln. The Property 2R Conjecture [GST] (the simplest remaining case of the Generalized Property R Conjecture) proposes that any link of two components in S3, on which some surgery yields (S1 × S2)#(S1 × S2), can be changed to the unlink by a series of moves called handle-slides. Let L denote the set of of all two-component links that contain the square knot Q and on which some surgery yields (S1×S2)#(S1×S2). It is argued in [GST] that there is an infinite family Ln ⊂ L of links that, for n ≥ 3 probably do not satisfy the Property 2R Conjecture, for if one of the family did, then a corresponding presentation of the trivial group could be trivialized by Andrews-Curtis moves, an outcome that seems very unlikely. On the other hand, it is also shown that each link in the family can be handle-slid to the unlink, once a single canceling Hopf pair is added. Thus all of the Ln are consistent with a Weak Generalized Property R Conjecture, a conjecture that would suffice to prove the smooth 4-dimensional Poincare ́ Conjecture for homotopy spheres without 1-handles [GST, Proposition 9.2]. There is a straightforward classification of L in [GST, Corollary 6.4], recounted here in Section 5: The classification assigns to each link a natural slope, indexed by those p q ∈ Q for which q is odd. But the clas-sification requires unspecified handle-slides of the other link component over Q and we were unable to resolve in [GST] how the family Ln fits into this classification. This left a puzzling gap ([GST, Question One]) between the 4-dimensional Kirby-calculus arguments (with 20-year-ol