Have you ever wondered if there is more to knots than meets your eye when tying your shoelaces? If so, here’s your answer: there’s a theory in mathematics called knot theory that delves into exactly this.
Study of closed curves
Knot theory is the study of closed curves in three dimensions and their possible deformations without one part cutting through another. Imagine a string that is interlaced and looped in any manner. If this string is then joined at the ends, then it is a knot.
The least number of crossings that occur even as a knot is moved in all possible ways denotes a knot’s complexity. This minimum number of crossings is called the order of the knot and the simplest possible knot has an order of three.
More crossings, more knots
As the order increases, the number of distinguishable knots increases manifold. While the number of knots with an order 13 is around 10,000, that number jumps to a million for an order of 16.
German mathematician Carl Friedrich Gauss took the first steps towards a mathematical theory of knots around 1800. The first attempt to systematically classify knots came in the second half of the 19th Century from Scottish mathematician-physicist Peter Guthrie Tait.
While the knot theory continued to develop for the next 100 years or so as a pure mathematical tool, it then started finding utility elsewhere as well. A breakthrough by New Zealand mathematician Vaughan Jones in 1984 allowed American mathematical physicist Edward Witten to discover a connection between knot theory and hyperbolic geometry. Jones, Witten, and Thurston all won the Fields medal, considered to be among the highest prizes for mathematics, for their contributions.
Many applications
These developments in the last few decades has meant that knot theory has found applications in biology, chemistry, mathematical physics, and even cosmology. Who knows, the possibilities with knots could possibly be endless.
Picture Credit : Google
