  Let $G$ and $H$ be disjoint embeddings of complete graphs $K_m$ and $K_n$ in
$\mathbb{R}^3$ such that some cycle in $G$ links a cycle in $H$ with non-zero
linking number. We say that $G$ and $H$ are *weakly linked* if the absolute
value of the linking number of any cycle in $G$ with a cycle in $H$ is $0$ or
$1$. Our main result is an algebraic characterisation of when a pair of
disjointly embedded complete graphs is weakly linked.
  As a step towards this result, we show that if $G$ and $H$ are weakly linked,
then each contains either a vertex common to all triangles linking the other or
a triangle which shares an edge with all triangles linking the other. All
families of weakly linked pairs of complete graphs are then characterised by
which of these two cases holds in each complete graph.
