Kitchen Sink, Gurgling, Slow Drain, Studor Vent

I have a 2 basin kitchen sink. It goes directly to a grease trap. The sink is vented underneath the cabinet via a sturdor vent and not externally. When draining, it gurgles a lot at first and then just becomes a really slow drain. If one side of the sink is draining, it will back up to the other side until it all drains. The drain lines are clear from any obstructions. Attached are pic of the drains under the sink. What' wrong????

Very straight forward clogged drain line. That is all it is. Snake the line, or if you already did your best, call a professional plumber or drain cleaner. Do not even think about using drain chemicals.

edit: I overlooked the "grease trap" part. I think johnjh20 has nailed it.

Drain lines are clear from any obstructions. Drains all the way to the grease trap have been snaked with out obstructions. When sink is attempting to drain, I can screw the studor vent and air seems to want to come out instead of in. It will drain without delay with the vent removed. But then the fumes become an issue under the cabinet.

To start out with, please don't take offense at the following.

You need to get a real plumber, who knows their trade, to take care of this.

From the description of your problem, and the picture, it is obvious to me that the original installer does not have a grasp of the fundamentals of correct drainage plumbing.

If the waste line goes into a grease trap, a sturdor vent will not work. Studor vents allow air into the system but not out. What is happening is when the water enters the grease trap it is trying to displace the air in the trap and the line. With no were to go it builds pressure preventing the water from draining. You need to remove the grease trap or run the vent through the roof.