I would try to elbow around the joist. Then get creative with a chase that will cover it all. The other option is to double up on one joist. Cut out offending section. Double up on blocking that sets on the wall and joins doubled up joists. If you are comfortable doing a minor electrical project forget about getting the city involved.

I'm not sure if you're intending to do this to 1 joist, or multiple joists.

you can box out 1 joist, supporting it by tieing the load into the neighboring joists. if you're planning on cutting a 6"-8" hole in multiple joists, there really is not way to do that.

Option 1: Move your range a couple of inches to the left or right, so you can send the pipe up between joists.

Option 2: Use a different kind of range hood--one where the whole thing goes all the way to the ceiling, or one you put cabinets over--so that you can send the pipe straight up between joists(i.e., off center relative to the hood and the range) without the off-centeredness being visible.

Option 3: Like Don says, elbow around the joist. Or, if this is just one joist you're proposing to cut, learn everything you can about the method Chrisk327 describes (or hire a reputable structural engineer to tell you what to do), and do that. But if you go that route, DO NOT jerry rig it, DO NOT just try to wing it or figure it out as you go--that joist is literally holding up the floor and/or roof above it, you don't want to get this wrong. Not to mention, an improperly supported joist can fall down into your kitchen... and fixing that ain't cheap.

Which of those options seems like the wisest course to you? Assuming #1 isn't possible--maybe the counters are in the way--I would go with #2. Having to put the range hood you have on eBay and get a different one is a lot cheaper and less time and hassle than having to take a crash course in residential structural engineering, buy the materials and tools, chop into a supporting beam of your house (and hope you don't cause permanent damage), and then fix the ceiling up again around it.

Leave the joist alone.
Have fabricated a transition which allows the duct to fit around the joist on either side, and another transition to connect above it and continue to the roof.

Chimney-style hoods are chosen because of the clean look, so building an offset or bulkhead will detract from the appearance. I know that you can safely "head-off" the intruding joist, but it involves doubling up the uncut joists at either side. This would call for removal of the ceiling (gulp). If it is actually a truss, you will need an engineer's stamped drawing as a "permission slip" to cut it. Do not cut a truss without an engineer's drawing; you will be in serious trouble. It would be far easier to design around this "existing condition".
Casey

Cutting one ceiling joist and installing headers each side of the vent might not require reinforcement of the trimmer joists but someone familiar with the building code (usually allows single trimmer joists to support a single header if within 3 ft of a support wall) would need to review the framing and if it is not allowed by the code you would probably need an engineer to stamp an alternative design. An engineer might be able to design a longer header above the ceiling joists or a vertical hanger from the roof structure above.

Thanks everyone for your ideas.....it is so tricky that I have decided to talk with the city permit office if it can be done legally in any way at all. Better pay for a permit than to have to tear it out or have a fire. There are actually double joists right in line with where the exhaust would go. I cannot move the stove as it is built in in a granite counter top. If the permit people tell me how it can be done I will hire the most experienced carpenter available. If I move the exhaust pipe in the ceiling forward from the wall to clear the joists, then the rangehood itself will extend way too far forward beyond the front of the stove. Any type of rangehood that I install would still need to go through the ceiling and then the roof except for an unducted one and that is unacceptable to me. Even if I went to an under the cabinet model which I could do as there is a 30 inch cabinet in place now which had a microwave under it, but.....even though if I could move the exhaust and the cabinet forward into the kitchen for clearance, the front of the rangehood will still extend too far beyond the front of my stove......and I am 5'9" and would have to duck all the time. Will keep working hard for a solution and will let you know.....Thanks again

How did you solve this? We are running into similar issue. There is a double joist blocking the vent and we were thinking of adding a box around to let the vent through.