Can we tell what dark matter surrounds a black hole by watching light bend?

Different dark matter distributions around supermassive black holes produce nearly identical gravitational lensing signatures — Einstein rings, image positions, magnifications all agree to within one part in a thousand. The one exception is the time delay between relativistic images, which amplifies the dark matter signal most strongly for the heaviest black holes. For M87* and Sgr A*, this means standard Event Horizon Telescope-style imaging probably can't distinguish dark matter models, but future time-domain observations might.