The common solutions to a system of polynomial equations in variables x, y, and z define an (algebraic) variety in 3-dimensional space. Two such varieties are said to be `linked’ if their union is a particularly nice kind of variety. Thus, linked varieties are in some sense complementary: one carries an imprint of the other.

Linkage has an algebraic incarnation that is used to study ideals. In the talk I will discuss how the idea that linked ideals are “complementary” can be used to unwind the detailed structure of a classification of ideals in the ring of polynomials in three variables. What that structure is only became clear to us—Oana Veliche, Jerzy Weyman, and myself—after extensive experimentation, and justifying it rigorously felt, at times, like solving a solitaire.