If you want to split a closed curve into 2 pieces, I often: 1. trim object with a line. delete the line. Then break object apart (Ctrl+K)
OR
2. select shape and use eraser tool to erase a fine line through the object. Then break object apart. (Ctrl+K)You can trying fooling around with the knife tool, but I use method 1 above Instead. The reason: I can make a precise curve before using it to trim into the shape. If you want to use knife anyway: some interesting tips: A: you can back-track on your knife drawing by pressiing shift key and going back to previous node. B. at end point of cut, press tab key to cycle through options for deleting resulting shapes.