Why Mathematicians Study Knots

Far from being an abstract mathematical curiosity, knot theory has driven many findings in math and beyond. Quanta magazine: Knot theory began as an attempt to understand the fundamental makeup of the universe. In 1867, when scientists were eagerly trying to figure out what could possibly account for all the different kinds of matter, the Scottish mathematician and physicist Peter Guthrie Tait showed his friend and compatriot Sir William Thomson his device for generating smoke rings. Thomson -- later to become Lord Kelvin (namesake of the temperature scale) -- was captivated by the rings' beguiling shapes, their stability and their interactions. His inspiration led him in a surprising direction: Perhaps, he thought, just as the smoke rings were vortices in the air, atoms were knotted vortex rings in the luminiferous ether, an invisible medium through which, physicists believed, light propagated. Although this Victorian-era idea may now sound ridiculous, it was not a frivolous investigation. This vortex theory had a lot to recommend it: The sheer diversity of knots, each slightly different, seemed to mirror the different properties of the many chemical elements. The stability of vortex rings might also provide the permanence that atoms required. Vortex theory gained traction in the scientific community and inspired Tait to begin tabulating all knots, creating what he hoped would be equivalent to a table of elements. Of course, atoms are not knots, and there is no ether. By the late 1880s Thomson was gradually abandoning his vortex theory, but by then Tait was captivated by the mathematical elegance of his knots, and he continued his tabulation project. In the process, he established the mathematical field of knot theory. We are all familiar with knots -- they keep shoes on our feet, boats secured to docks, and mountain climbers off the rocks below. But those knots are not exactly what mathematicians (including Tait) would call a knot. Although a tangled extension cord may appear knotted, it's always possible to disentangle it. To get a mathematical knot, you must plug together the free ends of the cord to form a closed loop. Because the strands of a knot are flexible like string, mathematicians view knot theory as a subfield of topology, the study of malleable shapes. Sometimes it is possible to untangle a knot so it becomes a simple circle, which we call the "unknot." But more often, untangling a knot is impossible.

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