This is a repost of an article I wrote in 2013, back when I was willing to take the time to explain physics stuff, and generate lots of images too.
It’s often said that topology is the branch of mathematics where they can’t tell the difference between a donut and a coffee mug. They each have a single hole (the mug’s handle and the donut hole), and that’s all that matters. If I may overanalyze this joke, the point seems to be that topology is so disconnected from our everyday experience. How is this useful?
I wish to explain one particular use of topology in physics: topological defects.
A topological defect is a sort of knot that exists in the microscopic structure of a material.* You can move the knot around from atom to atom, but you can’t untie it. We’ll get into how that works soon enough.
*Material is a vague term for “stuff”. Later I’ll discuss a few different materials including magnets, liquid crystals, and superconductors [Read more…]