Linkage in codimension three

The story of linkage in codimension two is well-understood, and was originally used to classify curves in $P^3$. For example, such a curve is in the linkage class of a complete intersection (licci) if and only if it is arithmetically Cohen-Macaulay. In codimension three (and beyond) only the forward implication remains true in general, but it is conjectured that if the Betti numbers are "small" then the reverse direction holds. I will share some ongoing joint work with Lorenzo Guerrieri and Jerzy Weyman regarding this topic, with an emphasis on examples. No prior knowledge of linkage is assumed.