A real sharp knife slides through rubber like through butter! But only under one special condition: the rubber that you want to cut should be drawn apart. This rubber should be under tension and never, never under compression. This is how it does not work, just try to cut as if rubber was wood or a potato. Then the rubber compresses around the edge of the knife. As rubber causes so much friction (yes, that's why it is used to keep cars on the road) the friction keeps the knife from cutting unless excessive force is used.
Therefore: lean in the inner ring of the tire and cut as it shown.