Crossed battery terminals and shorted out

Question edited for clarity.
Question moved to model category.

Probably took out the Engine Control Module. You need a good electrician, or shell out $259 on a guess. Or according to this, it fries the rectifier, (the white smoke you saw)

http://forums.sohc4.net/index.php?topic=64052.0

https://www.google.com/search?q=2007+triumph+1050+battery+connected+reverse+polarity

https://www.google.com/search?q=2007+Triumph+1050+ECM+repair

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Go to your battery find out where that smoke is coming from. It sounds like summer your wires melted together. That was cause everything the short out, make sure your battery is charged, take it out of the bike and charge it and make sure you got about 12.67 V coming out of it. When you tested smoke coming from anywhere as bad. You could have some wires melted together file your wire harness, make sure that's not so

If you jump started from a car you have fried the battery. Also possible cooked the rectifier/regulator unit. I suggest get a new battery and charge it at no more than 1 amp for a few hours. Fit the battery and see if the bike starts. If it does check the charge rate to the battery with a muti-imeter. You should be looking at just over 14 volts with about 5K RPM. Good luck and please remember never jump start from a car, van, lorry etc in fact don't jump start at all from anything.

Yes it sounds like one. OK, find your abttery terminals, one is marked in black cable and the other has a red sleeve near the terminal. Follow the red cabel to a box that contain a 30A fuse, see if this is blwon, which most likely. Replace fuse and now use a trickle charger as car batteries has toooo much amperage to jump start a bike. If you have another bike battery it is fine to jump start, never try using a car battery as many damages can happen. Once battery is fully charged ( normally in 8 hours) try again for your bike to start. Cheers. If good give a return comment please.

you must have crossed the battery cables. check all fuses under hood

Ck fuses and wire connections first. If no problem seen, get bike running and disconnect the ground cable @ battery. If bike dies the problem generally is in the stator or regulator assembly not keeping battery charged. Bike will run untill the battery is too low to keep the ignition system energized and the bike will die. If it continues running w/the ground disconnected, the battery could have a dead cell which will drain the battery in a very short time whether it is running or not.

to properly clean a carbarater

I drive the bike every day or two and it always starts immediately. Does this sound like a starter or solenoid or something else? Well tried to jump start the bike and it wouldn't start just the same clicking sound. Also tried to jump over the solenoid with a screwdriver and nothing happened. Then I tried to push start the bike and again nothing. I can tell the battery is weak but even push starting, I should be able to start it without a battery right? ,,,

When reversing polarity of the battery, which is what happens when jumper cables are reversed, this "charges" the chasis, which allows current to enter every electrical component through its ground pathway. This may cause component failure, commonly called" letting the magic smoke out". Since you say there is black smoke visible in the exhaust, the failure is in the injection area. Try to find a expert diesel tech to test all components which can effect the injection(sounds like a pulse-width modulation problem in the control circuit)