Brieskorn spheres bounding rational balls

Akbulut, Selman; Larson, Kyle

doi:10.1090/proc/13828

Brieskorn spheres bounding rational balls
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by Selman Akbulut and Kyle Larson PDF

Proc. Amer. Math. Soc. 146 (2018), 1817-1824 Request permission

Abstract:

Fintushel and Stern showed that the Brieskorn sphere $\Sigma (2,3,7)$ 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 $\Sigma (2,4n+1,12n+5)$ and $\Sigma (3,3n+1,12n+5)$ 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 $\Sigma (2,3,7)$. These handle diagrams necessarily contain 3-handles.

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