The knurled knob is the idle adjuster, and is indeed attached to the middle of the carbs. You will need to get the tank off to reattach it. After the seats are off, turn the fuel tap to off, and with a phillips driver remove the screw holding the fuel tap on and remove the fuel tap.
Prize off the fuel tap hose, there will be a little spillage, then undo and remove the front and rear tank bolts and lift the tank off.
Below the middle pair of carbs is where the idle adjust will screw back in, you'll need a torch.
Take a look at the big white connector under the front of the tank that goes to the ignition plug, the white wire in this plug can cause the problems so check its not corroding. Check the other plugs while your there.
Check the plugs coming up from the stator under the carbs for poor connections and melting, then check the black round connectors going to the voltage regulator under the front seat left side rear fairing for melting.
Hopefully you will find the problem in one of those plugs.
If you have problems starting the bike after it's just been running, try turning the fuel tap to OFF about Half mile before you stop, and turn it on again before you start the bike. If this cures the engine hot starting problem then you need to replace the carb float valves, as there worn and constantly letting fuel into the carbs and flooding them. This is a common problem when the bike has about 25,000 miles plus on it, sometimes less.