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On the $d$-invariants of Brieskorn spheres and $\mathrm{GL}(2,\mathbb{Z})$
We study the $d$-invariant from Heegaard Floer homology for Brieskorn spheres $Σ(p,q,r)$. Using surgeries on two singular fibers, we establish new relations of the $d$-invariants among various Brieskorn spheres. For $p=2$, we show that $d$ never vanishes if $q/r$ lies in certain intervals on the rat
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