Non-mitotic sets

We study the question of the existence of non-mitotic sets in NP. We show under various hypotheses that
• 1-tt-mitoticity and m-mitoticity differ on NP.
• T-autoreducibility and T-mitoticity differ on NP (this contrasts the situation in the recursion theoretic setting, where Ladner showed that autoreducibility and mitoticity coincide).
• 2-tt-autoreducibility does not imply weak 2-tt-mitoticity (from this it follows that autoreducibility and mitoticity are not equivalent for all reducibilities between 2-tt and T, although the notions coincide for m- and 1-tt-reducibility).