C. McA. Gordon's exceptional-slope bound conjecture

Let $M$ be a hyperbolic knot manifold, meaning a compact orientable irreducible 3-manifold whose boundary is a torus and whose interior admits a complete finite-volume hyperbolic structure. Let $\mathcal{E}(M)$ be the set of exceptional slopes, namely slopes $\alpha$ on $\partial M$ for which the Dehn filling $M(\alpha)$ is not hyperbolic, and let $\Delta(\mathcal{E}(M))$ denote the maximum distance between two slopes in $\mathcal{E}(M)$. The manifolds $M_1,M_2,M_3,M_4$ are the four specified hyperbolic knot exteriors obtained by Dehn filling the right-handed Whitehead link exterior.

C. McA. Gordon's conjecture. For any hyperbolic knot manifold $M$, we have $\#\mathcal{E}(M) \leq 10$ and $\Delta(\mathcal{E}(M)) \leq 8$. Moreover, if $M \ne M_1,M_2,M_3,M_4$, then $\#\mathcal{E}(M) \leq 7$ and $\Delta(\mathcal{E}(M)) \leq 5$.

The first bound is known in general, while the full strengthened statement remains open according to the supplied source context; results establish it in several cases, including when the first Betti number is at least two, and only finitely many manifolds can have exceptional-slope distance greater than five.