Why won't my insurance company pay for anesthesia services for my knee operation?

Out of network? It is unlikely the anesthesia would not be covered as a service for an indicated surgery but it is possible the anesthesiologist may have been out of network or didn't accept your insurance.
Call Them. This sounds like there might have been a coding error somewhere in the bill that was sent to the insurance company. No one would believe that you have to have a knee operation without anesthesia. I would call the facility where you had the procedure and ask them to help you with the insurance company. Good luck!