A mathematical knot is just like a regular knot, except that the ends are glued together. For decades, mathematicians have tried to categorize knots. That is, they want to list all the knots that there are. It is more difficult than you might think, because the different parts can move around, and it is hard to tell two knots apart. The next animated image illustrates this difficulty.
Knots can shift around.
Knots have become increasingly important to science. For instance, DNA molecules can be knotted. In the ideal math universe, knots are continuous curves that have no distinct parts. It is unknown how many inflexible parts you need to build most knots. A prize is also being offered.