Characterizing slopes for $5_2$

We prove that all rational slopes are characterizing for the knot $5_2$, except possibly for positive integers. Along the way, we classify the Dehn surgeries on knots in $S^3$ that produce the Brieskorn sphere $\mathrm{\Sigma }(2,3,11)$, and we study knots on which large integral surgeries are almost L-spaces.