As far as I know, 1-surgery on the figure-eight knot gives ($\pm$) the Brieskorn sphere $\Sigma(2,3,7)$. However, is there a citeable source for this? Sometimes Thurston's notes are mentioned, but I couldn't find a statement that the result of the surgery is this particular space.
2026-03-26 09:49:42.1774518582
1-surgery on the figure-eight knot: reference request
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I'm not sure what might be the earliest spot where it's mentioned, but you could refer Example 1.4 from Saveliev's book "Invariants of Homology 3-Spheres".