Brieskorn spheres bounding rational balls

Akbulut, S. and Larson, Kyle (2018) Brieskorn spheres bounding rational balls. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 146 (4). pp. 1817-1824. ISSN 0002-9939

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Abstract

Fintushel and Stern showed that the Brieskorn sphere Σ(2, 3, 7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls: the families Σ(2, 4n+1, 12n+5) and Σ(3, 3n+1, 12n+5) for n odd. We also provide handlebody diagrams for a rational homology ball containing a rationally slice disk for the figure-eight knot, as well as for a rational homology ball bounded by Σ(2, 3, 7). These handle diagrams necessarily contain 3-handles. © 2017 American Mathematical Society.

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