Venting kitchen sink below window

We have an unvented kitchen sink (old plumbing, never vented). The sink is below a window, and the nearest access for a vertical vent up is about 3-4 ft away horizontally.

I don't think we can get 6" above fixture rim due to the window height. Is the island venting approach the best? I have seen references to the "dirty arm" approach but am not sure exactly what the details would be. I also understand AAV, but would like to avoid it if possible.

We are in Illinois, and I will discuss the details of the approach with our plumbing inspector, but wanted to understand what the options are first.

Thanks very much!

You can run the drain and vent alongside the window then extend a dirty arm across to the sink.

Thanks very much for the quick reply.....

Can the dirty arm go 3-4' horizontally for a 2" pipe? This is a load bearing wall, so, I am also concerned about whether I can drill for the 2" horizontal run through standard 3 1/2" studs. It would have to pass through at least 3 studs.

A 1.5" dirty arm can be 42" long, and a 2" dirty arm can be 60" long.

Can you run the pipe beside the wall for a while (inside kitchen cabinets) instead of cutting studs?

Pretty sure sinks have been below windows for years... My kitchen sink just has a long trap arm before the tee...