Arithmeticity of hyperbolic link complements

Determine which hyperbolic link complements have arithmetic fundamental groups, extending the known classification of arithmetic hyperbolic knot complements and the arithmeticity result for the (2n)-chain links.

Background

The paper notes that Reid proved the figure-eight knot complement is the only hyperbolic knot complement with arithmetic fundamental group. It then contrasts this with link complements, many of which are related to quotients of hyperbolic 3-space by ideal, right-angled reflection groups.

For the (2n)-chain links, the cited theorem gives arithmeticity exactly for n=3 or 4. A broader classification for hyperbolic link complements remains unresolved, particularly beyond cases whose fundamental groups are themselves hyperbolic reflection groups.