I still don't get what you mean by knife edge. Are they Sliding? Are they braking away? If they are then you've found the edge of the tyre
... and as I asked above, have you tried different pressures as this is a whole lot cheaper than buying a new tyre. I found the Pilot Powers superb and very very confidence inspiring. Definitely no knife edge. They do however have a triangulated rear profile that means they fall to a set lean angle quickly and then take up a very stable lean. But this was right to the edge of the tyre. If you are trying to lean past that point then maybe you are just riding off the edge of the tyre. Or your rear tyre is over inflated.
C