[JimR]

Yes

it turns over and will start and run if I jump across the solenoid . It just won’t start with the key switch and the switch is new . Is works the headlights and shuts down the mower just won’t start it

You say you have power at the purple wire on the solenoid and that your lights work. Looking at the wiring diagram, that only leaves teh short section of black wire on the other side of the solenoid to be bad (or the solenoid itself). You've already replaced the solenoid. While the new one could still be bad, the odds are low.

I'd disconnect the black wire from the solenoid and throw a jumper wire in there from that terminal directly to the negative terminal on the battery. If it works correctly with that then you black wire is bad at one end or the other. If it doesn't then there is no other possibility but the solenoid being bad.