Lord Kelvin's idea that atoms were knots spurred the first knot tables
In the 1860s Lord Kelvin proposed that atoms were knots tied in the aether. That theory pushed Peter Guthrie Tait to build the first tables classifying knots, and the mathematics of closed, tangled loops has since reached DNA, quantum field theory and proposals for quantum computers.
A mathematical knot is a shoelace knot with its ends fused, so it can never be untied; the plain ring, called the unknot, is the simplest. Formally it is a circle embedded in three-dimensional space, and two knots count as the same if one can be pushed and stretched into the other without cutting the string or passing it through itself. Because a single knot can be drawn as countless different diagrams, the central question is telling whether two pictures show the same knot. An algorithm that always settles this exists, though its complexity is unknown, so in practice mathematicians compute invariants, quantities such as polynomials or groups that come out the same for every drawing of a given knot.
Vandermonde sketched a mathematical theory of knots in 1771, and Gauss later defined the linking integral. In 1885 Tait issued a catalogue of knots with at most ten crossings, along with the conjectures that bear his name, before the subject folded into topology. Early 20th-century workers such as Max Dehn and J. W. Alexander studied knot groups and the Alexander polynomial. In the late 1970s William Thurston brought in hyperbolic geometry, and Vaughan Jones's polynomial of 1984, extended by Witten, Kontsevich, Kauffman and others, linked knots to statistical mechanics and quantum field theory. Over six billion knots and links have now been catalogued.
Biology gave the field new work late in the 20th century. Knot theory can decide whether a molecule is chiral, and tangles, strands with both ends pinned, have helped explain how topoisomerase enzymes act on DNA. Extensions handle open curves, useful for proteins and real ropes, and higher-dimensional knots, where an n-dimensional sphere sits inside a space two dimensions larger. Topological quantum computation may rely on it too.
The human fascination is ancient. Archaeology shows knot-tying in prehistory, and knots appear in Chinese art from several centuries BC. Tibetan Buddhism uses the endless knot, and many cultures have used Borromean rings to stand for strength in unity. The monks who made the Book of Kells filled whole pages with Celtic knotwork.
Source: Knot theory
