On the Orbits of Computably Enumerable Sets
  The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, $\mathcal{E}$, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: The Scott rank of $\mathcal{E}$ is $\omega^{CK}_1+1; Not all orbits are elementarily definable; There is no arithmetic description of all orbits of $\mathcal{E}$; For all finite $\alpha \geq 9$, there is a properly $\Delta^0_\alpha$ orbit (from the proof).   April 6, 2007, minor changes   Nov 20, 2007, minor changes 
Peter Cholak, Rod Downey, Leo Harrington