  The reductions of an ideal $I$ give a natural pathway to the properties of
$I$, with the advantage of having fewer generators. In this paper we primarily
focus on a conjecture about the reduction exponent of links of a broad class of
primary ideals. The existence of an algebra structure on the Koszul and
Eagon-Northcott resolutions is the main tool for detailing the known cases of
the conjecture. In the last section we relate the conjecture to a formula
involving the length of the first Koszul homology modules of these ideals.
