Rereading the posts here, there seems to be some cross talking about exactly what is meant by groundfighting with a knife. IMHO, there are two distinct types of groundfighting with a weapon.
In the first case, an attacker comes at you with a weapon (knife for the purposes of this thread) drawn and the resultant struggles winds up going to the ground. In this case, if you are concentrating on a mount or dominant position, you are most likely already dead and don't know it. Realistically, if you haven't already at least partially isolated the weapon, your attacker is vigorously sticking into you repeatedly. For me, in this case I would have already tried to disarm my assailent before closing the distance and grappling. Assuming this didn't work, my entire effort would be to isolate the weapon and break something he needs to hold on to his weapon - ie his, fingers, wrist, elbow, or shoulder. Of course, at the same time, I'd need to protect myself from his "other weapons" like his fist and elbows, etc. I've had two encounters with a knife, one I avoided fighting entirely, one was a case like this. I got cut on both hands, but I lived. He went to the hospital and later to jail. (BTW- it's really hard to hold onto somebody when your hands are covered with blood)
The second type of ground fighting I've never encountered personally, but it involves grappling on the ground AND THEN your opponent pulls a knife. This requires a whole new set of strategies which, again IMHO, reside mainly in not allowing him to get that knife out. I honestly feel that I don't have the skills to handle the situation if he gets that knife to bear. It's too close and too hard to control. Unless I was able to disengage and get up, I feel that I would in all likelyhood die. Therefore, if I was able to recognize what he was trying to do - a BIG "if" - I would either attack his limb (break something) or get the hell off of him.