The Mathematics of Knots: From Sailor's Hitches to DNA Topology

The mathematical knot is not the knot you tie in your shoes. The shoe knot has loose ends; you can untie it by pulling. The mathematical knot is a closed loop, with the ends fused together, and the question is whether the loop can be deformed continuously into a simple unknotted circle without cutting it. Some loops can. Some cannot. Telling them apart turns out to be one of the deeper questions in topology, and the techniques developed to answer it have leaked into chemistry, biology, and quantum physics in ways that the founders of the field could not have anticipated.

Lord Kelvin's wrong idea that started everything

The field began with a mistake. In 1867, William Thomson, later Lord Kelvin, proposed that atoms were knotted vortices in the luminiferous ether. Different elements would correspond to different knots. The atomic spectrum, the periodicity of the elements, the chemistry: all of it would fall out of a complete classification of knots in three-dimensional space. It was a beautiful idea and almost entirely wrong: there is no ether, atoms are not vortices, and the periodic table has nothing to do with knot theory.

But Kelvin's wrong idea attracted Peter Guthrie Tait, a Scottish physicist who took the classification problem seriously. Tait, working alone and by hand, compiled the first systematic tables of knots, distinguishing them by the minimum number of crossings in any planar diagram. By the 1880s, Tait had tabulated all knots up to ten crossings, with corrections from Charles Kirkman and Mary Haseman extending the work. The tables stood, with refinements, for nearly a century.

What Tait did not have was a reliable way to prove that two knots with the same diagram were different. He had conjectures, some of which (the Tait conjectures) were not proved until the 1980s. He had intuition, which was usually right but occasionally wrong: a knot he had listed as different from another turned out to be the same in 1974, after a century of being separate entries in the table.

What an invariant is and why it matters

The fundamental tool of knot theory is the invariant: a property of a knot that does not change when the knot is deformed. Two knots can have the same invariant value while being different (so invariants are necessary but not sufficient), but if two knots have different invariant values, they must be different. Invariants reduce the question "are these knots the same" to the question "do they have the same invariants," which is often computable.

The simplest invariant is the minimum crossing number itself, but it is hard to compute and not very informative. The Alexander polynomial, introduced by James Alexander in 1928, was the first algebraic invariant: a polynomial in one variable that distinguishes most small knots. It is computable from a knot diagram by a procedure that takes some practice but is mechanical. The Alexander polynomial dominated the field until 1984, when Vaughan Jones, working on operator algebras for completely unrelated reasons, discovered the Jones polynomial.

The Jones polynomial is a polynomial in t (or q, depending on convention) that distinguishes knots more finely than the Alexander polynomial. It also has connections to physics that the Alexander polynomial does not: the Jones polynomial can be computed from a path integral in Chern-Simons gauge theory, as Edward Witten showed in 1989. The Jones polynomial of a knot is, in a precise mathematical sense, a quantity computable from a quantum field theory whose states are knotted configurations. This was so surprising that it led to a Fields Medal for Jones and changed how knot theory was understood.

The HOMFLY polynomial and the modern landscape

Within a year of the Jones polynomial, multiple groups discovered a generalization simultaneously: the HOMFLY polynomial, named for its discoverers Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter. HOMFLY is a polynomial in two variables that specializes to both the Alexander and Jones polynomials in different parameter limits. It distinguishes knots even more finely.

The Khovanov homology, introduced by Mikhail Khovanov in 1999, lifted the Jones polynomial to a homological invariant: instead of a polynomial, knots get a graded chain complex whose Euler characteristic is the Jones polynomial. Khovanov homology distinguishes knots that the Jones polynomial cannot, and connects knot theory to gauge theory and homological algebra in deep ways.

The current state of knot theory is that we have powerful invariants that distinguish almost all knots, but we do not have an algorithm that distinguishes all knots. The unknot recognition problem (given a knot diagram, is it the unknot) was shown to be in NP by Joel Hass, Jeffrey Lagarias, and Nicholas Pippenger in 1999. It was shown to be in coNP by Marc Lackenby in 2016. Whether it is in P remains open.

Knots in DNA

The unexpected application of knot theory came in molecular biology. DNA in cells is a long polymer that twists, folds, and crosses itself. When DNA replicates or transcribes, the strands become tangled, and the cell needs to untangle them. The enzymes that do this are called topoisomerases, and they cut, pass, and reseal DNA strands in ways that change the topology.

The discovery, made by James Wang and Nicholas Cozzarelli in the 1970s and 1980s, is that DNA in cells is not just topologically trivial: it can be knotted, and its knot type matters. Different topoisomerases produce different distributions of knots when acting on circular DNA, and the knot distributions can be analyzed using the same invariants developed by mathematicians a century earlier. The Jones polynomial of bacterial plasmids has been measured experimentally.

The biological picture is that the cell maintains DNA topology actively. There are enzymes that introduce knots, enzymes that resolve knots, and a steady-state distribution of topological types that depends on cell state. Cancer cells often have abnormal DNA topology, and topoisomerase inhibitors are an important class of chemotherapy drugs. The knot theory developed for hypothetical atoms in the ether turned out to describe the topology of life.

Knots in chemistry

Synthetic chemists have been able to construct molecular knots since the 1989 work of Jean-Pierre Sauvage, who shared the 2016 Nobel Prize in Chemistry partly for this. A molecular knot is a single molecule with a knotted topology, made by threading and connecting molecular components in a way that mechanically locks them. Trefoil knots, figure-eight knots, and more complex knots have all been synthesized.

The applications are speculative but interesting. Knotted molecules have unusual properties: they can be more rigid than their unknotted counterparts, more resistant to degradation, and have distinctive optical responses. The 2017 work of David Leigh's group at Manchester produced an 819 knot (a knot with eight crossings) made of 192 atoms, the most complex molecular knot yet synthesized. It was not clear what to do with it, beyond the fact that it existed.

Knots in quantum computing

The strangest current application is in topological quantum computing. Anyons are hypothetical quasiparticles in two-dimensional materials that, when braided around each other, transform the quantum state by a unitary operation. The braids are quantum knots, in a sense: their topology determines the quantum operation, and small perturbations of the braid that preserve its topology do not change the operation. This makes topological quantum computing inherently fault-tolerant, in principle.

The principle has been hard to realize. Microsoft's Station Q has worked on Majorana fermions in superconductor-semiconductor heterostructures for over a decade, with mixed results. Other approaches use the fractional quantum Hall effect or topological superconductors. The state of the field is that topological quantum computing is theoretically attractive and experimentally elusive, with the relevant anyon physics requiring extreme cold and clean materials.

If topological quantum computing ever works, the knot invariants developed by Tait, Alexander, Jones, and Khovanov will be the language in which quantum algorithms are described. Knot diagrams will be circuit diagrams. The Jones polynomial of a particular knot will be the answer to a quantum computation. The mathematical structure that started as a wrong theory of atoms will, by a long detour, end up describing the computers that might replace silicon.

What sailors knew

The deeper irony is that sailors knew about the most useful knots long before mathematicians studied them. The bowline, the clove hitch, the sheet bend: these are practical inventions, refined over centuries of working with rope under load. The mathematical theory does not improve them, and the sailor's knot does not need a polynomial to be useful.

What the mathematics does is connect domains that had no obvious connection: rope and DNA, sailing and quantum computing, vortices in ether (still wrong) and topological order in condensed matter (newly right). The theory of knots is a small example of a recurring pattern in mathematics: a structure studied for one reason turns out to describe phenomena in domains that were not part of the original motivation. Sometimes the original motivation was wrong. The structure is real anyway.