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Brieskorn spheres bounding rational balls


Authors: Selman Akbulut and Kyle Larson
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 57R65; Secondary 57M99
DOI: https://doi.org/10.1090/proc/13828
Published electronically: October 30, 2017
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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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Additional Information

Selman Akbulut
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
Email: akbulut@math.msu.edu

Kyle Larson
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48823
Email: larson@math.msu.edu

DOI: https://doi.org/10.1090/proc/13828
Received by editor(s): April 27, 2017
Received by editor(s) in revised form: May 16, 2017
Published electronically: October 30, 2017
Communicated by: David Futer
Article copyright: © Copyright 2017 American Mathematical Society