Riding Lawn mowers - My lawn mower will not start.

My riding lawn mower will not turn over.
The battery was bad, I replaced it with a new one and also replaced the solenoid.

My riding lawn mower will not turn over.
The battery was bad, I replaced it with a new one and also replaced the solenoid.
The engine will turn over if the solenoid terminals are jumped but the engine will not fire.
There is power to the starter and power to the switch. I tried checking the switch with a meter and it seems to be ok.
Any suggestions?

Ok, riding mowers often have a few interlock circuits for safety. These can prevent cranking, and shut down the ignition. The safety circuits are normally attached to the seat, the clutch/brake, and the mowing deck. When you try to start it you should be sitting in the seat. Most mowers have a seat sensor switch, where the mower will not start if you are not in the seat. It may even have a setup where the engine shuts off if you get out of the seat. This is to shut the engine down in the case of a roll over, or accident.
The clutch/brake: most mowers have the clutch, and the brake built into one pedal, while others have no clutch, and only have a brake. Regardless the brake pedal has to be pressed hard down until it stops. This is because there is normally a start inhibit switch that will prevent the mower from cranking unless the pedal is pressed all the way down.
The mowing deck: normally has a safety switch built in that prevent the mower from cranking if the deck is switched to the "ON" position. The deck may even have a circuit to kill the engine if you are not on the seat when the deck is switched on.
So, hop on the seat, fully depress the brake, turn the mowing deck off, and see if it will crank and start. If it will not start then you may have problems with the wiring. You may have other problems with the engine as well.

I have already gone through those steps:
I sat on the seat, I pressed the brake down as far as possible and nothing happened. I cut the switch out under the seat and wired the wire together. The engine still would not turn over.

It sounds like you have two problems if you checked everything in my last post. The reason I mentioned this again is because 95% of the time this is what the problem is, and it is SUPER easy to over look.
You said you jumped the wires under the seat, sometimes they have to be open to make the mower operate, dependent on how it is wired.
Unplug all wires from the engine except the starter. Jump the starter solenoid to crank the engine. The engine should start if it is OK. The way the mower wiring kills the engine is to ground the primary circuit in the magneto. If you unplug all the wiring except the starter, this will prevent the mower from being able to kill the spark to the engine, and it should run when cranked: if there is nothing wrong with the engine. If the engine starts you will have to ground the magneto primary wire, or the plug wires to kill the engine. You can also pull a plug wire but that can shock the lights out of you, so use a very well insulated tool.
As far as the cranking with the key goes, the solenoid should be receiving voltage from the key to trigger it to crank. The small electrical terminal on the solenoid triggers it to crank when you send 12 volts to it. So when you turn the key to crank, it should send 12 volts to the solenoid. Also remember that the solenoid will not work if it is not grounded well.