Before revision:  In particular, for the case of k=2, we improve the current best lower bounds on biplanar crossing numbers by a factor of 1.37 for complete graphs, and by a factor of 1.34 for complete bipartite graphs. We extend our results to the k-planar crossing number of complete (bipartite ) graphs, for any positive integer k \geq 2.
After revision:  In particular, for the case of k=2, we improve the current best lower bounds on biplanar crossing numbers by a factor of 1.37 for complete graphs, we provide a new approximation factor of 3.17, improving over the best previous factor of 4.34. We provide similar improved approximation factors for the k-planar crossing number of complete (bipartite ) graphs, for any positive integer k \geq 2.