Battery fully charged all systems work but negative jumper cable 13+ volts between positive cable and battery negative post

You don't say what happens when you try to start the machine. Make/Model/Year would also help with diagnoses. However, in general you have three possibilities (assuming the wiring is good).

1. The solenoid (starter relay) is receiving current from the key switch but is not engaging properly or consistently. To check this component, bypass it. To bypass it use one side of a jumper cable. Clip to the positive of the battery with one end of the jumper cable (red). Hold to the large input wire on the starter with the other end of the same jumper cable (red). This should cause the starter to turn over strongly. If so, replace the solenoid.

2. If you try the above scenario 3 to 6 times and find the starter turns over inconsistently, or at times does not turn over at all, you may have flat spots on the armature of the starter(or worn brushes). In this case you will have to replace the starter (this is the least likely scenario).

3. BEFORE you consider either of the above, consider the battery. This is an easy check if you have a simple volt meter. It's much better to have an old fashioned needle meter rather than a digital. The needle is more sensitive, much quicker, and makes a clearer diagnostic tool. But not to worry, a digital still works (sometimes the results are not as clear).

1. Set the meter to DC Volts
2. Attach the red and black leads of the meter to the positive and negative posts of the battery. Most modern day meters don't care if the polarity is correct or not. However, if you have a needle meter, best to put the positive of the battery with the red wire of the meter; otherwise the needle will not be happy with you.
3. Note voltage. It should read above 13 volts if the battery is good and fully charged. If it reads below 12 volts it needs charged or it has bad cells. If after charging a few hours, the voltage is still below 12 volts then the cells are bad, replace it.
4. Crank the engine while keeping your eyes on the meter. The meter should not fall much below 11 Volts. If it falls below 10 volts or worse yet below 9 volts, the battery has a bad cell or two. Replace the battery, charge it, and repeat the test.
5. If the solenoid just rapidly clicks watch the meter needle.
   1. If the meter needle rapidly jumps up and down below 10 volts, back to 13, below 10, back to 13 then it's more likely you have a bad starter.
   2. If the needle simply seems to vibrate a little you more likely have a bad solenoid. Refer to steps 1 and 2.
6. If the engine starts, rev it up and watch the meter.
   1. If the charging system is working the voltage on the meter should quickly rise above 12 or 13 volts.
   2. If it rises up strongly towards the 14 volt range this indicates the charging system of the machine is working.
   3. If it plays around down near 12 volts you are reading the recovered voltage of a good battery, but the charging system is not working.
   4. If it simply stays below 11 volts, the battery and the charging system are both suspect. Replace the battery first. Fully charge it, and then repeat the tests before worrying about the charging system.

I can't suggest any parts numbers because you did not include the very important Make/Model/Year. However, these general instructions will work for most riding lawn and garden tractors.

In these cases you have three possibilities (assuming the wiring is good).

1. The solenoid (starter relay) is receiving current from the key switch but is not engaging properly or consistently. To check this component, bypass it. To bypass it use one side of a jumper cable. Clip to the positive of the battery with one end of the jumper cable (red). Hold to the large input wire on the starter with the other end of the same jumper cable (red). This should cause the starter to turn over strongly. If so, replace the solenoid.

2. If you try the above scenario 3 to 6 times and find the starter turns over inconsistently, or at times does not turn over at all, you may have flat spots on the starter. In this case you will have to replace the starter (this is the least likely scenario).

3. BEFORE you consider either of the above, consider the battery. This is an easy check if you have a simple volt meter. It's much better to have an old fashioned needle meter rather than a digital. The needle is more sensitive, much quicker, and makes a clearer diagnostic tool. But not to worry, a digital still works (sometimes the results are not as clear).

1. Set the meter to DC Volts
2. Attach the red and black leads of the meter to the positive and negative posts of the battery. Most modern day meters don't care if the polarity is correct or not. However, if you have a needle meter, best to put the positive of the battery with the red wire of the meter; otherwise the needle will not be happy with you.
3. Note voltage. It should read above 13 volts if the battery is good and fully charged.
4. Crank the engine while keeping your eyes on the meter. The meter should not fall much below 11 Volts. If it falls below 10 volts or worse yet below 9 volts, the battery has a bad cell or two. Replace the battery, charge it, and repeat the test.
5. If the solenoid just rapidly clicks watch the meter needle.
   1. If the meter needle rapidly jumps up and down below 10 volts, back to 13, below 10, back to 13 then it's more likely you have a bad starter.
   2. If the needle simply seems to vibrate a little you more likely have a bad solenoid. Refer to steps 1 and 2.
6. If the engine starts, rev it up and watch the meter.
   1. If the charging system is working the voltage on the meter should quickly rise above 12 or 13 volts.
   2. If it rises up strongly towards the 14 volt range this indicates the charging system of the machine is working.
   3. If it plays around down near 12 volts you are reading the recovered voltage of a good battery, but the charging system is not working.
   4. If it simply stays below 11 volts, the battery and the charging system are both suspect. Replace the battery first. Fully charge it, and then repeat the tests before worrying about the charging system.

I can't suggest any parts numbers because you did not include the very important Make/Model/Year. However, these general instructions will work for most riding lawn and garden tractors.

After assuring battery is charged and all electrical connections are clean and tight, you can try this,
If your tractor has the external solenoid (separate from the starter), you can by pass this solenoid by jump starting the starter with booster cables.
Negative cable will go on neg post of battery to a good ground on engine and then positive cable on positive post of battery, then just touch the other positive end to positive starter post (where the large cable is connected from solenoid). If the starter engages, chances are the the solenoid is bad or could be ignition switch is bad.

To check your charging system, get a "multimeter"
They're available for under $10

Set the meter to D.C. Voltage, and hook to the battery terminals, Positive to Positive, Negative to Negative.

Start bike and read the meter.

If it's working right, you should be getting about 13-13.5 volts at idle, and it should rise slightly when you rev the engine.

Battery full charge, start fuse good, valves adjusted correct

Low battery, bad starter or bad cable to starter

IF STARTER RELAY BAD THE STARTER WONT MAKE NO NOISE AT ALL HEADLIGHTS WONT DIM.IF YOU CLICKING NOISE BATTERY VOLTAGE TOO LOW. MAKE SURE BATTERY VOLTS ARE 12.6 VOLTS AND USE TEST LIGHT TO SEE IF STARTER GETTING POWER.COULD HAVE BLOWED IGNITION FUSE OR FAULTY IGNITION SWITCH. TO CHECK STARTER SOLENOID FIRST TURN OFF RADIO THEN REMOVE NEGATIVE BATTERY CABLE AND POSITIVE BATTERY CABLE.THEN REMOVE STARTER BUY A REMOTE SWITCH.YOU NEED BATTERY JUMPER CABLES HOOK BATTERY JUMPER CABLES TO BATTERY PUT POSITIVE CABLE ON POSITIVE BATTERY POST THEN PUT NEGATIVE BATTERY CABLE ON NEGATIVE POST. HOOK BATTERY CABLES TO STARTER USE NEGATIVE CABLE GROUND STARTER BODY AND PUT POSITIVE BATTERY CABLE ON THE LARGE POSITIVE POST ON STARTER SOLENOID. NOW TAKE REMOTE SWITCH HOOK RED WIRE TO STARTER SOLENOID POSITIVE POST THEN HOOK BLACK WIRE TO REMOTE SWITCH TO STARTER SOLENOID SWITCH TERMINAL WITH LETTER S THEN PRESS REMOTE IF STARTER SPINS STRONG WITH A GOOD KICK STARTER GOOD.YOUR PROBLEM COULD BE CORRODED WORN BATTERY CABLES. WEAK BATTERY FAULTY IGNITION SWITCH. IF STARTER DONT WORK WHEN YOU PRESS THE REMOTE SWITCH STARTER SOLNOID BAD OR STARTER ITSELF BAD.IF ALL IS GOOD. CODE SCAN CAR FOR THEFT SECURITY FAULT CODE.

It should be positive to positive, negative to negative - parallel, not serial.

The simple way to tell is by how many cables you have. If you've got one positive and one negative, and only one jumper, it's a 24 volt system and should be connected in serial. If you've got one positive and one negative and two jumpers, it's a 12 volt system, and should be connected parallel.

If in doubt, connect up in parallel and test things. If everything works, it's a 12 volt system and you're good to go. If everything doesn't work (lights are dim, inverter doesn't give AC, etc.), it's a 24 volt system and you need to change to serial.

Serial - positive to system connection on battery 1, negative on battery 1 to positive on battery 2, negative on battery 2 to system negative.

Parallel - positive on battery 1 connects to positive on battery 2 and the system positive, negative on battery 1 connects to negative on battery 2 and the system negative.

Using a voltmeter, check that voltage is identical to battery volts at the other end of cables. Check across the battery terminals, across connectors on battery, between negative post on battery and connection at starter, between positive post and end of negative cable then from postiive post and block. Wherever you find a variation, clean and re-check reading.If this does not help, there are sub system teststs you can do as well but I believe you will likely locate your problem before getting into that.