I am struggling to find the crossing number of the complete bipartite graph $K_{4,4}$. The best range I can get to is $\text{cr}(K_{4,4}) \leq 16$ (obtained by putting all vertices onto a regular octagon), which is still messy of course.
Any help is appreciated!
We see that $cr(K_{4,4})\leq4$ by placing the partite sets in a way that they are normal to each other.
Next, if the crossing number is less than 4, removing some 3 edges from $K_{4,4}$ would result in a planar graph. Now, by Euler's formula ($v-e+f=2$), we conclude that this new graph has $7$ faces. Since the original graph was bipartite, every face must be bounded by at least $4$ edges (bipartite graphs have only even cycles). Therefore, the sum of bound degree $\sigma \geq 28$. At the same time, $\sigma \leq 2\cdot 13=26$, a contradiction. Thus, $cr(K_{4,4}) = 4$.