Around 1958, Hill described how to draw the complete graph $$K_n$$ K n with $$\begin{aligned} Z(n) :=\frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \end{aligned}$$ Z ( n ) : = 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ crossings, and conjectured that the crossing number $${{\mathrm{cr}}}(K_{n})$$ cr ( K n ) of $$K_n$$ K n is exactly $$Z(n)$$ Z ( n ) . This is also known as Guy’s conjecture as he later popularized it. Towards the end of the century, substantially different drawings of $$K_{n}$$ K n with $$Z(n)$$ Z ( n ) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line $$\ell $$ ℓ (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by $$\ell $$ ℓ . The 2-page crossing number of $$K_{n} $$ K n , denoted by $$\nu _{2}(K_{n})$$ ν 2 ( K n ) , is the minimum number of crossings determined by a 2-page book drawing of $$K_{n}$$ K n . Since $${{\mathrm{cr}}}(K_{n}) \le \nu _{2}(K_{n})$$ cr ( K n ) ≤ ν 2 ( K n ) and $$\nu _{2}(K_{n}) \le Z(n)$$ ν 2 ( K n ) ≤ Z ( n ) , a natural step towards Hill’s Conjecture is the weaker conjecture $$\nu _{2}(K_{n}) = Z(n)$$ ν 2 ( K n ) = Z ( n ) , popularized by Vrt’o. In this paper we develop a new technique to investigate crossings in drawings of $$K_{n}$$ K n , and use it to prove that $$\nu _{2}(K_{n}) = Z(n) $$ ν 2 ( K n ) = Z ( n ) . To this end, we extend the inherent geometric definition of $$k$$ k -edges for finite sets of points in the plane to topological drawings of $$K_{n}$$ K n . We also introduce the concept of $${\le }{\le }k$$ ≤ ≤ k -edges as a useful generalization of $${\le }k$$ ≤ k -edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of $$K_{n}$$ K n in terms of its number of $${\le }k$$ ≤ k -edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of $$K_{n}$$ K n and show that, up to equivalence, they are unique for $$n$$ n even, but that there exist an exponential number of non homeomorphic such drawings for $$n$$ n odd.