jeremy,

Mose ESCs have in them what's called a BEC (Battery Eliminator Circuit) which allows the flight battery to supply voltage to both the motor and receiver, so no receiver battery is needed.

The 'battery wire' coming from the ESC is dual purpose...

First, it supplies voltage to the receiver and servos just as a receiver battery would.
Second, it sends the throttle signal from the receiver to the ESC.

So, the only connection you need is from the ESC to the throttle channel of your receiver, which in your case, is channel 3.