Volumes of chain links

TL;DR

This paper proves that for sufficiently large n, the minimally twisted n-chain links are not the smallest volume hyperbolic manifolds with n cusps, using a combination of formulas, computer computations, and theoretical bounds.

Contribution

It provides the first rigorous proofs for cases where n is at least 12, 60, and for certain twist parameters, establishing when chain links are not minimal volume.

Findings

For n ≥ 60, chain links are not minimal volume.

Computer-assisted proofs for 12 ≤ n ≤ 25.

Certain twist configurations cannot be minimal volume for large n or large |m|.

Abstract

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|.