Fill a plane with oscillators, one at every point, each carrying a phase — an arrow on its own little clock face. If the coupling is doing its job, neighbors mostly agree, and the field of arrows looks like a well-combed lawn. But there is one kind of disturbance that combing cannot fix: a point the arrows circulate around. Walk a small loop around such a point and the phase you carry turns through one full 2π — the loop has a winding number. You drew exactly this integer with your own hand in the topology module; here it stops being a game and becomes a physical object. A point the field winds around is called a vortex, and every claim in this module — and the wall the whole library is built on — is about what a population of these costs and does.
The figure below is a field of phase arrows (hue underneath repeats the same information, since phase is a color wheel as naturally as it is an angle). Click to drop a vortex with winding +1; toggle the button to drop −1 instead (an anti-vortex — the arrows circulate the other way). The readout measures the winding of the phase carried around the border of the whole box: it always equals the sum of the charges you placed inside. Individual arrows change everywhere with every vortex you add — but that one integer is doing bookkeeping no local rearrangement can cheat.
Why call this a defect rather than just a messy spot? Because of what happens when you let the field relax. Relaxation is the coupling doing what coupling does: every arrow turns, a little at a time, toward the average of its neighbors. Smooth ripples iron out and their energy drains toward zero. A vortex does not. The arrows near it can shuffle, the core can shift, but the winding is an integer counted around any loop — and an integer cannot change by a sequence of tiny moves. To remove it you would have to tear the field somewhere, which coupling will not do. This is topological protection: the same reason your drawn loop in the topology module could not unwind without crossing the hole.
There is exactly one way a vortex can die, and it is the way that matters for everything in the next section: find an anti-vortex. A +1 and a −1 close together look, from far away, like nothing at all — walk a big loop around the pair and the windings cancel; the far field barely knows they exist. Drag them together and they annihilate outright, leaving a combed lawn. Drag them apart and you must pay for every step: the field between them stays torqued. That cost of separation is the entire subject of section 02.