Unless you're cutting the apples against a glass cutting board or something similar, that shouldn't dull the knife at all. And one piece of rope shouldn't do anything significant either.
I'm beginning to think that the steel on that knife is very soft. Either that or you're still getting a wire edge that is rolling over as soon as you use it just a little bit.
When you're using the lansky system, make sure that the stone is moving towards the edge of the blade during the sharpening process--as if the knife is carving the stone. That should help a little with preventing a wire edge. Sometimes if I have a knife that is really prone to getting a wire edge, I will finish up with one very light pass over each side of the edge with a coarser stone to pull off any wire edge that may have formed--again, moving the stone toward the knife edge as if carving the stone with the knife.
One other thing that can be important is that I find it important to alternate strokes on the knife, especially toward the very end of the sharpening process, turning the knife over so that the stone alternates sides on the edge with each stroke. This also helps prevent a wire edge.