  A knot K in the 3-sphere is said to have Property nR if, whenever K is a
component of an n-component link L and some integral surgery on L produces the
connected sum of n copies of S^1 x S^2, there is a sequence of handle slides on
L that converts L into a 0-framed unlink. The Generalized Property R Conjecture
is that all knots have Property nR for all n.
  The simplest plausible counterexample could be the square knot. Exploiting
the remarkable symmetry of the square knot, we characterize all two-component
links that contain it and which surger to S^1 x S^2 # S^1 x S^2. We argue that
at least one such link probably cannot be reduced to the unlink by a series of
handle-slides, so the square knot probably does not have Property 2R. This
example is based on a classic construction of the first author.
  On the other hand, the square knot may well satisfy a somewhat weaker
property, which is still useful in 4-manifold theory. For the weaker property,
copies of canceling Hopf pairs may be added to the link before the handle
slides and then removed after the handle slides. Beyond our specific example,
we discuss the mechanics of how addition and later removal of a Hopf pair can
be used to reduce the complexity of a surgery description.
