Volumes of chain links

Agol has conjectured that minimally twisted n-chain links are the smallest volume hyperbolic manifolds with n cusps, for n at most 10. In his thesis, Venzke mentions that these cannot be smallest volume for n at least 11, but does not provide a proof. In this paper, we give a proof of Venzke's statement for a number of cases. For n at least 60 we use a formula from work of Futer, Kalfagianni, and Purcell to obtain a lower bound for volume. The proof for n between 12 and 25 inclusive uses a rigorous computer computation that follows methods of Moser and Milley. Finally, we prove that the n-chain link with 2m or 2m+1 half-twists cannot be the minimal volume hyperbolic manifold with n cusps, provided n is at least 60 or |m| is at least 8, and we give computational data indicating this remains true for smaller n and |m|.