Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly 
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor.$$

In the literature, two general constructions of drawings of $K_n$ with $Z(n)$ crossings appear: 

1) the *cylindrical* (or *tin can*) drawing, where vertices are placed on the boundaries of the bottom and the top circular face of a cylinder and edges are drawn as geodesics,

2) a *$2$-page* (or *cycle*) drawing where the vertices form a regular $n$-gon, with the diagonals that are "more horizontal than vertical" drawn inside the $n$-gon and the remaining diagonals drawn outside the $n$-gon. Recently Abrego et al. showed that all optimal $2$-page drawings of $K_n$ are basically the same (up to some boundary effects for odd $n$).

**The question:**

>Are there other known classes of drawings of $K_n$ with $Z(n)$ crossings? I am especially interested in explicit constructions like the two above.


References:

[B. M. Abrego, O. Aichholzer, S. Fernandez-Merchant, P. Ramos, and G. Salazar, The 2-page crossing number of $K_n$, 2012, arXiv:1206.5669](http://arxiv.org/abs/1206.5669)

[L. Beineke and R. Wilson, The early history of the brick factory problem, The Mathematical Intelligencer 32(2) (2010), 41--48](http://dx.doi.org/10.1007/s00283-009-9120-4)

[H. Harborth, Special numbers of crossings for complete graphs, Discrete Mathematics 244 (2002), 95--102](http://www.sciencedirect.com/science/article/pii/S0012365X01000784)

[F. Harary and A. Hill, On the number of crossings in a complete graph, 
Proc. Edinburgh Math. Soc. (2) 13 (1963), 333--338](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=3473660)

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**Edit:**

Apparently there is a rather broad class of drawings with crossing number $Z(n)+O(n^3)$, which generalize the cylindrical drawings:

3) A *spherical* drawing is a drawing on the sphere where edges are drawn as shortest arcs.

Moon showed that a random spherical drawing of $K_n$ has expected crossing number $\frac{1}{64} n(n-1)(n-2)(n-3)$.

[J. W. Moon, On the Distribution of Crossings in Random Complete Graphs, J. Soc. Indust. Appl. Math. 13 (1965), 506--510](http://www.jstor.org/stable/2946445)

I think the following construction must be known but I couldn't find any reference.

For $n$ even, if one places $n/2$ pairs of antipodal points on the sphere (so that no three pairs are on the same great circle), then the crossing number of the induced spherical drawing is $Z(n)+X(n)$ where the term $X(n)$ denotes the number of crossings of the $n/2$ arcs (half-circles) connecting the pairs of antipodal points (so $X(n) < n^2/8$). For the spherical analogue of cylindrical drawings, we have $X(n)=0$.

**Question 2:**

>>Is there some simple criterion for the positions of the pairs of antipodal vertices so that the half-circles can be drawn in a non-crossing way? Can one obtain, in this way, "antipodal" spherical drawings with $Z(n)$ crossings that are not cylindrical?

**Question 3:**
>>Dropping the condition that vertices should form antipodal pairs, are there spherical drawings with $Z(n)$ crossings that are not cylindrical?