The Chicago Loop is an alternate way of venting the trap installed on an under counter island sink or other similar applications where a conventional vertical vent stack is not feasible.
As with all drains, ventilation must be provided to allow the flowing waste water to displace the sewer gas in the drain and then the vacuum which would otherwise form as the water flows down the pipe. A Chicago Loop provides an elegant solution: when the drain is opened, water displaces the air to the tee. The water will flow down while the air is displaced upward to the vent. After the water has passed, air will be replenished through the vent. The key to a functional Chicago Loop is that the top elbow must be as close to, if not higher than, the "flood level" (peak possible drain water level) as possible. This ensures the vent will never become waterlogged. The second way provides a simpler path for the return airflow after the water has passed the drain.
The cost of installation is high because of the number of elbows and small pieces of pipe required; with modern plastic drain pipes, the largest outlay will be labor. Street elbows will be helpful. Alternatively, installation of an air admittance valve can be used in some cases to keep costs down.
Alternately if moving sink to an island sink, install the P-trap below the floor of the island and vent off the top of the drain. Attach toward the trap and reverse 180 degrees so any water in the vent flows down the drain. Slope drain down, slope vent up, and attach to existing vent from previous existing fixture that is now abandoned. Patch previously existing drain to become vent. In Canada, the national plumbing code requires that the minimum trap arm be at least 2 times the pipe diameter, (I.E 1.25 inch trap needs a 2.5 inch trap arm, 1.5 needs a 3 inch pipe etc.) and that the vent pipe be one size larger than the drain that it serves, also a cleanout is required on both the vent and the drain. The reason for this is in the event of a plugged sink, the waste water will back up and go down the vent, possibly plugging the vent (as it is under the countertop)for that reason a clean-out would permit the cleaning of the pipes.