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.
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 3, improving over the best previously known approximation factor of 4.03. For complete graphs, and by a factor of 1.34 for complete bipartite graphs.