LAUSR

It is an open problem whether Kirk’s σ-invariant is the complete obstruction to a link map f : S+2 ∪ S −2 → S4 being link homotopic to the trivial link. The link homotopy invariant associates to such a link map f a pair σ(f) = (σ+(f),σ−(f)), and we write σ = (σ+,σ−). With the objective of constructing counterexamples, Li proposed a link homotopy invariant ω = (ω+,ω−) such that ω± is defined on the kernel of σ± and which also obstructs link null-homotopy. We show that, when defined, the invariant ω± is determined by σ∓, and is strictly weaker. In particular, this implies that if a link map f has σ(f) = (0, 0), then after a link homotopy the self-intersections of f(S+2) may be equipped with framed, immersed Whitney disks in S4∖f(S −2) whose interiors are disjoint from f(S+2).