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Braid groups have many important practical applications. For example, they are used to construct efficient and robust public key encryption algorithms.7 Another promising direction is designing quantum computers based on creating complex braids of quantum particles known as anyons. Their trajectories weave around each other, and their overlaps are used to build “logic gates” of the quantum computer.8 There are also applications in biology. Given a braid with n threads, we can number the nails on the two plates from 1 to n from left to right. Then, connect the ends of the threads attached to the nails with the same number on the two plates. This will create what mathematicians call a “link”: a union of loops weaving around each other. In the example shown on this picture, there is only one loop. Mathematicians’ name for it is “knot.” In general, there will be several closed threads. The mathematical theory of links and knots is used in biology: for example, to study bindings of DNA and enzymes.9 We view a DNA molecule as one thread, and the enzyme molecule as another thread. It turns out that when they bind together, highly non-trivial knotting between them may occur, which may alter the DNA. The way they entangle is therefore of great importance. It turns out that the mathematical study of the resulting links sheds new light on the mechanisms of recombination of DNA. In mathematics, braids are also important because of their geometric interpretation. To explain it, consider all possible collections of n points on the plane. We will assume that the points are distinct; that is, for any two points, their positions on the plane must be different. Let’s choose one such collection; namely, n points arranged on a straight line, with the same distance between neighboring points. Think of each point as a little bug. As we turn on the music, these bugs come alive and start moving on the plane. If we view the time as the vertical direction, then the trajectory of each bug will look like a thread. If the positions of the bugs on the plane are distinct at all times – that is, if we assume that the bugs don’t collide – then these threads will never intersect. While the music is playing, they can move around each other, just like the threads of a braid. However, we demand that when we stop the music after a fixed period of time, the bugs must align on a straight line in the same way as at the beginning, but each bug is allowed to end up in a position initially occupied by another bug. Then their collective path will look like a braid with n threads. Thus, braids with n threads may be viewed as paths in the space of collections of n distinct points on the plane.10
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