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Weak Property 2R and Generalized Square Knots

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Alexander Zupan (Nebraska)

In the late 1980s, David Gabai proved the Property R Conjecture, which asserts that the only Dehn surgery (the process of removing a solid torus and gluing it in a different way) on the 3-sphere that yields the product manifold S^2 x S^1 is the obvious one -- namely, 0-surgery on the unknot. This talk concerns those 2-component links L in the 3-sphere with a Dehn surgery to the connected sum of two copies of S^2 x S^1. Such a link can be viewed as a blueprint for building a 4-manifold X, which is homotopy equivalent to the standard 4-sphere, and the Generalized Property R Conjecture states that all such links can be standardized by natural operations. We prove that when one component of L is the connected sum of a torus knot and its mirror image (a generalized square knot), L satisfies the Weak Generalized Property R Conjecture, and the corresponding 4-manifold X is diffeomorphic to the standard 4-sphere. This is joint work with Jeffrey Meier.