The triviality of a certain second-order invariant of link homotopy

A link map is a map of spheres into another sphere with pairwise disjoint images, and a link homotopy is a homotopy through link maps. It is conjectured that link maps of two 2-spheres in the 4-sphere are classified, up to link homotopy, by an invariant due to Kirk. A "second-order" invariant was proposed by Li to detect counterexamples to this conjecture, but while his (published) examples were later found to be in error, it remained an open question as to whether counterexamples could be detected in this manner. In this talk I will discuss the (very geometric) constructions of these invariants, and outline a proof that Li's invariant cannot detect such examples; indeed, it is a strictly weaker invariant.