I reversed the poles trying to jump start my 2011 troy built bronco mower.

With a new battery I get nothing. What have I apparently blown? A fuse or as relay ?

Could be a fuse or fusible link. Look over the owners manual for a wiring diagram.

More than likely you have blown the fuse. Follow the red/positive cable from the battery. It will go to the solenoid. The fuse will be within a few inches of the solenoid.