Extending drawings of complete graphs into arrangements of pseudocircles

Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for "pseudolinear" drawings of $K_n$, we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\mathbb{S}^2$ in which the vertices of $G$ are represented as points -- no three on a great circle -- and the edges of $G$ are shortest-arcs in $\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $\gamma_e$ such that the only vertices in $\gamma_e$ are the ends of $e$; (2) if $e\ne f$, then $\gamma_e\cap\gamma_f$ has precisely two crossings; and (3) if $e\ne f$, then $e$ intersects $\gamma_f$ at most once, either in a crossing or an end of $e$. We use Properties (1)--(3) to define a pseudospherical drawing of $G$. Our main result is that, for the complete graph, Properties (1)--(3) are equivalent to the same three properties but with "precisely two crossings" in (2) replaced by "at most two crossings". The proof requires a result in the geometric transversal theory of arrangements of pseudocircles. This is proved using the surprising result that the absence of special arcs ( coherent spirals) in an arrangement of simple closed curves characterizes the fact that any two curves in the arrangement have at most two crossings. Our studies provide the necessary ideas for exhibiting a drawing of $K_{10}$ that has no extension to an arrangement of pseudocircles and a drawing of $K_9$ that does extend to an arrangement of pseudocircles, but no such extension has all pairs of pseudocircles crossing twice.