You can discharge into a kitchen sink from the top like you would into a laundry tub and you would be fine; its done with portable units all of the time since you still meet the air gap requirement.
You can connect into your existing sink drain between the sink trap adapter and the p-trap, providing that its all 2" piping for the trap arm (I can still get away with 1.5" piping for a traditional clothes washer, but the newer style units require 2" piping as they discharge like commercial units). But this type of installation is really ill advised; it is done with a laundry tub because the laundry tub is deep enough to handle any backup, but doing it with a kitchen sink would not only put you at risk of flooding if the basins filled up, but you would put your dishwasher at risk of backflow, and at the very least you would definitely get a mound of suds backing up into your sink basins every time the washing machine discharged as well as contaminating your sink with filthy laundry water. And even if you were able to squeeze in another fitting for your laundry standpipe connection, you would still have to meet the standpipe requirements.
Your best bet would be to cut open the wall in your sink cabinetry, and install a tee below your sink trap arm in the kitchen sink stack and run a wet vented trap arm out to the standpipe (if wet venting is allowed where you are).
But you really should install and vent a stand pipe properly into a proper wash box behind where the laundry unit is going to go. Like everyone else is telling you, it will cost you less to do it properly in the first place than it will to pay for repairs and grief later on.