A surgery is complicated.  There are lots of possible parties involved.  This includes the surgeon and anesthesiologist, but also the nurses, CRNA, surgical techs, and the assistant surgeon.  When a mistake is made during a procedure, each party is liable for their own errors due to New York medical malpractice.  Indeed, New York has not adopted a captain of the ship doctrine where the surgeon is liable for all surgical errors.  Each party will have their own fault.