The "trace" as in the bit with the hook on it, doesn't go through a swivel on a pulley rig. The 'backbone' of the pulley rig (often made from trace material) does. the trace itself just clips on to one end of the pulley rig backbone.
If your line is getting cut by the swivel in the middle of the pulley rig, you have several options. Heavier line Swivels with thicker material in the eyes. Proper 'pulley' clips which have a roller in them.