We find the minimal number of non-trivial links in an embedding of any complete kk-partite graph on 7 vertices (including K7, which has at least 21 non-trivial links). We give either exact values or upper and lower bounds for the minimal number of non-trivial links for all complete kk-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.
Fleming, T. and B. Mellor, 2009: Counting links in complete graphs, Osaka J. Math. 46, 173--201, http://projecteuclid.org/euclid.ojm/1235574043.