thinking from the perspective of an attorney - unless the contractor asked and received permission from the architect or engineer of record to cut the joist, it his responsibility (to repair or prove it is structurally acceptable) and the liability is his. By advising him of a repair method, you will likely assume a large share of his liability. I would advise doing nothing more than requiring contractor to submit his plan for the repair, or his analysis which shows the joist is structurally sound - for your review: or put it back like it was and change the duct alignment to accomodate the truss without cutting.