theres only 2 paths for the circuit. In a general sense, these 2 paths are signal and ground. You can see signal as +(positive) and ground as -(negative)
so when you connect all the + together(and taking consideration that nothing goes wrong), of course it will be connected physically. Same for the - side(aka ground)
and to check all the points that are suppoosed to be connected, whether are those electrically connected. You can use a multi meter, set to continuity mode, use the 2 leads from multi meter and check for continuity. If the points thats supposed to be connected all nicely soldered, you will hear a sound from the multi meter to indicate those points are electrically connected.
anyway, lastly, heres something interesting for LP wiring. For normal lp wiring(aka no fancy push pull or extra switches for coil tap or in phase/out of phase etc), theres actually 2 general ways(of course with geetar wiring, there can be more, but for easy sake, i just mentioned the general 2).
First way, when the pickup selector in in center position, where the bridge and neck pickup are both connected in parallel, turning either the bridge volume or neck volume pot, will affect both pickup volume
2nd way, again, same thing, when pickup selector in center position, both pickup will be connected in parallel. But let say if we are adjusting the bridge volume pot, only the bridge pickup volume will be affected. The neck pickup volume wont be affected. Similarly, if we are only adjusting the neck pickup volume pot, only the neck pickup volume be affected.
anyway, the difference are in the way how the pots input/output are wired. If you want to know more, just go google for les paul wiring with individual volume control or les paul modern wiring/50 wiring. Theres plenty info and diagram out there for it.
have fun..
