Can you get hiv if you sit on a public toilet seat and you have a cut on your butt?

Nope: No risk at all. You can't get HIV without sex or sharing injection equipment with an infected person. If exposures like this could transmit the virus, there would be at least occasional infected patients who had none of the usual risk factors. But even the busiest HIV/AIDS clinics and practices simply never see such persons. That's how we know such exposures carry no risk. Don't worry about it.