A knot, in mathematical parlance, is a closed loop of string that may be tangled with itself. The simplest knot is the unknot — a plain circle, untwisted. The next simplest is the trefoil knot, which looks like the loop in a pretzel: three crossings, three lobes. After that comes the figure-eight knot, then knots with five, six, seven and more crossings, organised by their minimal crossing number.
A fundamental question of knot theory is: given two knots, are they the same? Two knots are considered the same if one can be continuously deformed into the other without cutting the string. Determining whether two knot diagrams represent the same knot is harder than it sounds — the same knot can look completely different from one drawing to the next.
The most powerful tool for this question is the knot polynomial: an algebraic invariant that assigns each knot a polynomial in such a way that equivalent knots get the same polynomial. If you compute the polynomials of two knots and they disagree, the knots are definitely different. If they agree, the knots might be the same — but you have at least narrowed the search down.
This article is about how knot polynomials work, why they exist, and what they have taught mathematicians about the deep connections between topology, algebra, statistical mechanics, and quantum field theory.
Two knots, two diagrams
The picture below shows two knots: the unknot (a plain circle, on the left) and the trefoil (the simplest non-trivial knot, on the right). To the eye, they look obviously different. To rigorous knot theory, the difference has to be proved — there has to be a way of showing that no sequence of continuous deformations turns one into the other.
The unknot’s Jones polynomial is, by convention, . The trefoil’s Jones polynomial is . Two different polynomials, two different knots — proved.
What makes this work is that the Jones polynomial is, by construction, invariant under continuous deformation of the knot. No matter how you redraw the trefoil — stretching, wiggling, repositioning the crossings — the polynomial stays . The trefoil can never be turned into a polynomial- unknot, so the two knots are genuinely distinct. The whole machinery of knot theory rests on finding such invariants.
Reidemeister moves: the rules of the game
Two diagrams represent the same knot if and only if they can be transformed into each other by a sequence of three local operations, called the Reidemeister moves:
- R1: Add or remove a “kink” — a small twist that doesn’t change the knot.
- R2: Push one strand under or over another, creating or removing two crossings of opposite type.
- R3: Slide a strand under a crossing, rearranging three local strands.
Kurt Reidemeister proved in 1927 that these three moves are sufficient: any two diagrams of the same knot are related by some finite sequence of moves. The trouble is that the sequence might be very long, and there is no efficient algorithm in general for finding it. So checking whether two knots are equivalent by searching through Reidemeister moves is not, in general, feasible.
A knot invariant is a function on knot diagrams that doesn’t change under Reidemeister moves. Once you have one, you can compute it on each of two diagrams. If the invariants disagree, the diagrams represent different knots. If they agree, they might represent the same knot — depending on how strong the invariant is.
The Alexander polynomial (1923)
The first algorithmically computable knot invariant was the Alexander polynomial, introduced by James Waddell Alexander II in 1923. To compute it, you label each “region” of a knot diagram (the connected pieces the diagram cuts the plane into), set up a matrix of region-versus-crossing relations, and take the determinant. The result is a polynomial in a single variable that doesn’t change under Reidemeister moves.
The Alexander polynomial of the trefoil is . The figure-eight knot’s polynomial is . The two are different, so the polynomial distinguishes them.
But the Alexander polynomial has a notable weakness: it cannot distinguish a knot from its mirror image. The Alexander polynomial of the right-handed trefoil and the left-handed trefoil are both , even though the two knots are genuinely different (a fact that turned out to be subtle to prove). For sixty years, this was a stubborn limitation.
The Jones polynomial (1984)
In 1984, Vaughan Jones — a New Zealand-born operator algebraist working on a completely unrelated problem about von Neumann algebras — accidentally discovered a new knot invariant. The polynomial Jones found is not the same as Alexander’s; it is computed in a fundamentally different way, using a recursive formula related to so-called skein relations that express the polynomial of any knot in terms of the polynomials of simpler knots.
The Jones polynomial satisfies a beautiful recursion: at any crossing of a knot diagram, write for the diagrams obtained by leaving the crossing alone, switching the over- and under-strands, or smoothing the crossing into two strands. Then
Together with the boundary condition , this recurrence determines the polynomial of any knot. The trefoil works out to , the mirror trefoil to . The Jones polynomial distinguishes a knot from its mirror image in many cases where the Alexander polynomial fails. It is genuinely stronger.
Jones’s discovery was a sensation. The Jones polynomial brought together knot theory and operator algebras, and a year later Witten and others realised that it had a natural interpretation in Chern–Simons quantum field theory, opening a flood of connections between topology and quantum physics. Jones was awarded the Fields Medal in 1990 for the work.
The HOMFLY polynomial and beyond
In 1985, six mathematicians (Hoste, Ocneanu, Millett, Freyd, Lickorish, Yetter — hence “HOMFLY”) independently discovered a two-variable polynomial that generalises both Alexander’s and Jones’s. The HOMFLY polynomial recovers the Alexander polynomial at one specialisation of its variables and the Jones polynomial at another. It is strictly stronger than either, distinguishing pairs of knots that both Alexander and Jones fail to separate.
Stronger invariants kept coming. Khovanov homology (1999) is a vast generalisation: instead of assigning each knot a polynomial, it assigns a chain complex whose Euler characteristic recovers the Jones polynomial. Two knots are distinguished if their Khovanov homologies are different, and Khovanov homology is strictly stronger than the Jones polynomial. Knot Floer homology, developed by Ozsváth and Szabó starting in 2003, is another such “categorified” invariant. Whether any such finite-data invariant can distinguish every pair of distinct knots — the so-called knot recognition problem — is still open.
Why anyone cares
Beyond the intrinsic mathematical interest, knot theory has applications that have grown in surprising directions.
Biology. DNA molecules in living cells are long, double-stranded, and can become knotted as they are compressed into the small space of a cell nucleus. Specific enzymes called topoisomerases can cut, untangle, and reseal DNA — a process essential to cell division. The mathematical knot theory of how DNA can be knotted, and what kinds of unknotting are biologically possible, is now a standard tool in molecular biology.
Statistical mechanics. Partition functions of certain two-dimensional lattice models — the Potts model, the six-vertex model — turn out to compute knot polynomials when the lattice is woven into a knot pattern. This is no coincidence; it is the discovery that drove Jones to recognise the connection between his operator-algebra invariant and knot theory.
Quantum field theory. Witten’s 1989 paper showing that the Jones polynomial arises from a quantum Chern–Simons theory was one of the deepest applications of quantum field theory to topology, foreshadowing the development of topological quantum field theory (TQFT) as a general framework. The mathematics of knot polynomials thus connects directly to quantum physics in a way that nobody anticipated before 1984.
Quantum computing. Topological quantum computers, proposed by Alexei Kitaev, would store information in knot-like braiding patterns of quasi-particles called anyons. The Jones polynomial controls the resulting computations, and proposed implementations rely on the same algebraic structures Jones discovered.
A discipline reborn
Knot theory began in the 19th century as a curiosity of recreational mathematics. Tait, Kelvin, and others tried to classify knots as part of a (now-defunct) atomic theory; their work survived as the foundation of a small mathematical subject that grew slowly through the early 20th century.
Then Vaughan Jones, working on something else entirely, opened a door in 1984. Behind it lay connections to operator algebras, quantum field theory, statistical mechanics, molecular biology, and quantum computing — all of which had been waiting, unrecognised, for the right algebraic invariant to come along. The Jones polynomial transformed knot theory into one of the most active and connected subjects in modern mathematics, and its tendrils continue to spread.
A polynomial that distinguishes two pieces of tangled string can tell you, with the right interpretation, about the structure of DNA, the behaviour of two-dimensional magnetic systems, the partition functions of quantum field theories, and the possible architectures of fault-tolerant quantum computers. That single piece of algebra carries that much content is one of the more remarkable facts in late-twentieth-century mathematics.
Frequently asked
Who invented knot polynomials?
James Waddell Alexander II discovered the first knot polynomial in 1923 — a polynomial in one variable now called the Alexander polynomial. Vaughan Jones discovered a second knot polynomial in 1984 while studying von Neumann algebras in operator algebras; the connection to knots came as a complete surprise even to him. The Jones polynomial turned out to be vastly more powerful than the Alexander polynomial, and it earned Jones the 1990 Fields Medal. The HOMFLY polynomial (discovered independently by six mathematicians in 1985) generalises both.
Why is the knot problem hard?
Because the same knot can be drawn in infinitely many ways. Knot diagrams can be deformed by 'Reidemeister moves' — three basic ways of changing a diagram without changing the underlying knot. Two diagrams represent the same knot if and only if they can be transformed into each other by Reidemeister moves, but checking this directly would require an exhaustive search through an infinite space of possible move sequences. Knot polynomials sidestep this entirely: they assign each knot a polynomial that doesn't change under Reidemeister moves, so two knots with different polynomials are definitely different.
Can two different knots have the same polynomial?
Yes — even the powerful Jones polynomial does not distinguish all knots. The Alexander polynomial fails to distinguish the trefoil from its mirror image (giving the same polynomial). The Jones polynomial distinguishes those two but fails on other pairs. There is, surprisingly, still no known knot polynomial that distinguishes every pair of distinct knots, although several stronger invariants (Khovanov homology, knot Floer homology) come close. Whether *any* finite-data invariant can distinguish all knots is a deep open question.
Why do knots show up in mathematics anyway?
Because they are the simplest non-trivial example of one-dimensional objects embedded in three dimensions, and they appear naturally in many contexts. In biology, the long DNA molecules in cells can knot themselves and must be unknotted by enzymes called topoisomerases. In statistical mechanics, partition functions of certain lattice models compute knot polynomials. In quantum field theory, Edward Witten showed in 1989 that the Jones polynomial arises naturally from a Chern–Simons quantum field theory, making knots central to the geometric study of quantum theories.