Irrationality conjecture for the Chern–Simons invariant of a hyperbolic surgery

Let $5_2^*$ be the mirror of the knot $5_2$ in Rolfsen’s knot table, and let $r_s$ be the invariant used to study the $\mathit{SU}(2)$-Chern–Simons functional. Irrationality conjecture. The value

$$r_s\left(S^3_{1/2}(5_2^*)\right)$$

is an irrational number. The paper computes this value numerically, with error at most $10^{-50}$, and notes that no 3-manifold is known whose $\mathit{SU}(2)$-Chern–Simons functional has an irrational critical value. Whether such a manifold exists remains open.