Reading hidden topology in light, even when energy leaks away

When I explain topology to students, I start with a knot in a rope. You can stretch it, twist it or shake it, but the knot stays until you cut the rope. Physicists have found that some materials and devices carry similar "knots" in how waves move through them. These are topological properties, labeled by whole numbers that don't change under small imperfections. That robustness is why topology has become one of the central ideas in modern physics. It promises electronics, photonics and quantum devices that tolerate defects and noise.