Ok so here is where everyone gets a chance to jump on me because I am always saying how much I hate AAV's and that I will fire anybody working for me that puts one in............with one exception; and the island vent is that exception. I truely feel that I would rather have an AAV there than a loop vent, and the reason why is that even though clean out's must be installed on the horizontal leg (your diagram does not show them but code does require it) if this vent plugs nobody will ever know that it is plugged. I think that an AAV makes more sense here than that convoluted mess of unnecessary fittings. However, many times all of this mess can be cured by simply upsizing the drain lateral to 3" and essentially stack venting the sink into the 3" If I remember correctly as I don't have a copy of the code handy, you can go something like 13' with 3" before hitting a vent. Please correct me if the distance is off.