Mathematical Study of Knots
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Introduction
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.
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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.
Three simple knots (from top left, clockwise): the unknot, the trefoil and the square knot.
Figures by Merrill Sherman/Quanta Magazine; source: David S. Richeson
Knots can also combine to form new knots. For example, combining a simple knot known as the trefoil with its mirror image produces a square knot. (And if you join two identical trefoil knots, you make a granny knot.)
Join a trefoil knot and its mirror image to make a square knot.
Using terminology from the world of numbers, mathematicians say the trefoil is a prime knot, the square knot is composite and, like the number 1, the unknot is neither. This analogy was further supported in 1949 when Horst Schubert proved that every knot either is prime or can be decomposed uniquely into prime knots.
Another way to create new knots is to intertwine two or more knots, forming a link. The Borromean rings, so named because they appear on the coat of arms for the Italian House of Borromeo, are a simple example.
To form the Borromean rings, three rings are interlinked, even though no two individual circles are linked.
Thomson and Tate were not the first to view knots in a mathematical way. As early as 1794, Carl Friedrich Gauss wrote about and drew examples of knots in his personal notebook. And Gauss’ student Johann Listing wrote about knots in his 1847 monograph Vorstudien zur Topologie ("Preliminary Studies of Topology") — which is also the origin of the term topology.
But Tait was the first scholar to work on what became the fundamental problem in knot theory: the classification and tabulation of all possible knots. Through years of painstaking work using only his geometric intuition, he found and classified all prime knots that, when projected onto a plane, have at most seven crossings.
The unknot and all prime knots with seven or fewer crossings (ignoring mirror images) arranged in the style of the periodic table.
In the late 19th century, Tait learned that two other people — the Rev. Thomas Kirkman and the American mathematician Charles Little — were also studying this problem. With their combined efforts, they classified all prime knots with up to 10 crossings and many of those with 11 crossings. Amazingly, their tables up to 10 were complete: They didn’t miss any knots.
It’s remarkable that Tait, Kirkman and Little accomplished so much without the theorems and techniques that would be discovered in the years to come. But one thing that worked in their favor was the fact that most small knots are "alternating," meaning they have a projection in which the crossings exhibit a consistent over-under-over-under pattern.
Alternating knots have properties that make them easier to classify than nonalternating knots. For example, finding the minimum number of crossings for any projection of a knot is difficult. But Tait, who for years mistakenly assumed all knots were alternating, conjectured a way to tell if you’ve found that minimum number: If an alternating projection has no crossings that can be removed by flipping over part of the knot, then it must be the projection with the minimum number of crossings.
Tait called any crossing that can be removed by flipping over part of the knot "nugatory," or inconsequential.
This and two more of Tait’s conjectures about alternating knots ended up being true. Yet these famous conjectures were not proved until the late 1980s and early ’90s using a mathematical tool developed in 1984 by Vaughan Jones, who won the Fields Medal for his work in knot theory.
Unfortunately, alternating knots only take you so far. Once we get into knots with eight or more crossings, the number of nonalternating knots grows quickly, making Tait’s techniques less useful.
This eight-crossing knot, drawn as the true-lover’s knot, cannot be drawn with an alternating projection.
The original table of all 10-crossing knots was complete, but Tait, Kirkman and Little double-counted. It wasn’t until the 1970s that Kenneth Perko, a lawyer who had studied knot theory at Princeton, noticed that two of the knots are mirror images of each other. They are now known as the Perko pair in his honor.
These two 10-crossing knots, known as the Perko pair, are the same knot.
Over the last century, mathematicians have found many clever ways to determine if knots are truly different. Essentially, the idea is to identify an invariant — a property, quantity or algebraic entity that is associated with the knot and can often be computed simply. (These properties have names like colorability, bridge number or writhe.) Armed with these labels, mathematicians can now easily compare two knots: If they differ in any given attribute, then they aren’t the same knot. None of these properties, however, is what mathematicians call a complete invariant, meaning that two different knots may have the same property.
Because of all this complexity, it may be no surprise that the tabulation of knots is still ongoing. Most recently, in 2020, Benjamin Burton classified all prime knots up to 19 crossings (of which there are almost 300 million).
Traditional knot theory makes sense only in three dimensions: In two dimensions only the unknot is possible, and in four dimensions the extra room allows knots to untie themselves, so every knot is the same as the unknot.
However, in four-dimensional space we can knot spheres. To get a sense of what this means, imagine slicing an ordinary sphere at regular intervals. Doing so yields circles, like lines of latitude. However, if we had an extra dimension, we could knot the sphere so the slices, now three-dimensional rather than two, could be knots.
When we slice a sphere in three-dimensional space, we get circles. But slices of a knotted sphere in four-dimensional space could be knots.
This idea was behind one of the biggest recent results in knot theory. In 2018, then-graduate student Lisa Piccirillo settled a 50-year-old question about an 11-crossing knot first discovered by John Conway. The question had to do with a property called sliceness. As we’ve seen, when we slice a knotted sphere in four dimensions, we obtain a knot or link in three dimensions. Sometimes we can obtain a given knot from a nice smoothly knotted sphere, but for other knots the sphere has to be knotted and crinkled up like a piece of wastepaper. Piccirillo proved, in essence, that Conway’s knot was of the latter type. In technical lingo, she proved that it is not "smoothly slice."
The Conway knot, which Lisa Piccirillo proved is not smoothly slice.
Knot theory has crisscrossed the mathematical landscape over the centuries. It began as an applied area of mathematics, with Thomson attempting to use knots to understand the makeup of matter. As that idea faded, it became an area of pure mathematics, a branch of the intriguing and still unpractical domain of topology. But in recent years knot theory has again become an applied area of mathematics, as scientists use ideas from knot theory to investigate fluid dynamics, electrodynamics, knotted molecules such as DNA and so on. Fortunately, while scientists were busy studying other things, mathematicians were building catalogs of knots and the tools to untangle their secrets.
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