Optimal crossing-free Hamiltonian circuit drawings of Kn

The number of crossing-free Hamiltonian circuits in planar drawings of K_n is studied. In particular, it is shown that for planar drawings of K_n, (1) there are drawings having as many as $\text{(}\text{3}\text{20}\text{)·10}{}^{\mathrm{<mtext></mtext><mtext>n</mtext><mtext>3</mtext><mtext>]</mtext>}}$ such circuits and (2) no drawing contains more than $\text{2·6}{}^{\mathrm{n?2}}\text{(}\text{n}\text{2}\text{])!}$