Not unless you want to create backups for your kitchen sink.Can I drain the tub into the existing vent from the kitchen sink below instead of draining it into the main drain/vent stack?
Any water coming down the pipe, will either pull the trap seal from the kitchen sink p-trap, creating bad smells,
Or overfill the sink.
Vents pipes are not normally even run with waste fittings, knowing that only air move through them.
The tub could be tied in below the kitchen sink drain, if you run a new waste line and tie it in below the kitchen sink.