20 hp 42" deck will not turn over. Will start if I

Yes. Verify it is the battery by jumping mower off with car battery. If turns over, replace battery. If cannot turnover by jumping, try bypassing start solenoid by tapping positive cable to battery post on starter. Starter should turn. If starter doesn't turn, have starter tested.

Make sure belt is installed like drawing.

Yes or a tight valve. Also have had faulty afterfire solenoids cause this

With the key in the run position, using a jumper cable, jump from the positive post on battery to terminal on starter motor. If it runs you have a solenoid or ignition switch problem. (assuming all electrical connections are clean). If it doesn't crank the battery is the culprit.

Can you hear the Solenoid click when to try to start? If not, I had the same problem it was the Solenoid. Sears insisted it must be the battery even though it was weak but not dead. Replaced the battery but it still would not work. Replaced the Solenoid and did the trick.

If the solenoid just clicks, that usually indicates a low battery voltage. But if starts by jumping the solenoid, as you say, then that indicates the solenoid has burnt contacts inside itself. Some are rebuildable, mostly just replaceable. I would still, first, disconnect the battery, remove the cables from the solenoid and clean both the wires and the contact points of the solenoid before discarding the solenoid. I would rule out all safety switches at that point because it is trying to work. If it was a safety issue, the solenoid would not recieve power to ingage (or click).

if it restarts immediately, check the vent in the gas cap for restriction. also, the solenoid on the bottom of the carburetor may be sticking or getting weak. if you have to let it sit for awhile before it retarts, the ignition module may be breaking down when it heats up. i'd bet on the gas cap or solenoid though

Hi, W/D here.
Try cleaning all of the contacts at the solenoid, and the wires that attach to it. That will probably get you going.

--W/D--

ck the terminals battery, solenoid. are they clean? and tight?

check and see if the white wire on the starter solenoid has voltage when the key is in the start position. If so, the solenoid is bad. If not, back probe wire until you find voltage.