Do you see that flat tab between the 2 big posts, that is where the trigger wire goes, and i believe it's that purple wire with the white stripe in your picture,as its that way on other John Deere mowers.
when you turn the key to start, the trigger wire gets energized, creates a magnetic field in the solenoid, which pulls the solenoid in to make contact between the 2 big posts, as well as it meshes the starter gear to the flywheel gear.