Do you have a gray tank, and is it vented separately through another vent pipe? If not, it should be. If it is vented via another vent pipe, then you can use an air-admittance valve under the counter to vent the kitchen sink. This is a under the counter one-way valve that will let air into the drain system, but not let sewer gas back out.
The purpose of the sink vents is to allow air in to run behind the water so the drain drains properly. The tank vent allows air to escape the tank when water flows in, as well as allowing air into the tank while you drain it. Another function of the tank vent is to allow sewer gasses to escape. Typically, those are stinky, even for a gray tank, so you really want the tank vent going up and out the roof.