Knot Theory: Telling Tangles Apart for Real

A knot is a closed loop, not a piece of string

In everyday language a 'knot' has loose ends you tie and untie. In mathematics, a knot is a closed loop embedded in three-dimensional space — imagine tangling a piece of string and then fusing its two ends together so there's no way to simply pull it free. The central question of knot theory is deceptively simple to state and often brutally hard to answer: given two tangled loops, are they secretly the same knot, meaning one can be continuously deformed into the other without ever cutting the string, or are they genuinely different?

Reidemeister moves: the only moves allowed

Kurt Reidemeister proved in 1927 that any continuous deformation of a knot diagram (a 2D projection with over/under crossings marked) can be broken down into a sequence of exactly three local moves, now called Reidemeister moves:

Type I    — twist or untwist a single loop (adds/removes one crossing)
Type II   — slide one strand completely over/under another (adds/removes two crossings)
Type III  — slide a strand across a crossing of two other strands (no change in crossing count)

Any two diagrams of the same knot are connected by a finite sequence of these three moves, and conversely two diagrams connected by Reidemeister moves always represent the same knot. This turns 'is this a trefoil or an unknot in disguise?' from a vague topological question into, in principle, a finite search — though for complicated tangles that search can still be enormous.

Crossing number and writhe

The simplest invariant is the crossing number: the minimum number of crossings in any diagram of the knot. The unknot (a simple loop with no real tangling) has crossing number 0; the trefoil, the simplest true knot, has crossing number 3; the figure-eight knot has crossing number 4. Crossing number is a genuine topological invariant — it doesn't change no matter how you wiggle the knot — but it's expensive to compute in general, since you'd need to prove no simpler diagram exists.

A related but different quantity is writhe: the signed sum of crossings in a specific diagram, where each crossing counts as +1 or −1 depending on the strands' relative orientation. Unlike crossing number, writhe is not a topological invariant of the knot itself — it changes under Type I Reidemeister moves — but it is invariant under Types II and III, which makes it useful in contexts like DNA supercoiling, where the actual physical diagram (not just the abstract knot type) matters.

The Alexander polynomial: an invariant that actually distinguishes knots

Crossing number alone can't always tell two knots apart, so knot theorists build stronger invariants — algebraic objects computed from a diagram that provably don't change under any Reidemeister move. The Alexander polynomial (James Alexander, 1928) was the first practical one: a Laurent polynomial Δ(t) computed from a matrix built out of the knot diagram's crossings, such that equivalent diagrams always yield the same polynomial (up to a harmless factor of ±tⁿ).

unknot:            Δ(t) = 1
trefoil (3_1):      Δ(t) = t − 1 + t⁻¹
figure-eight (4_1):  Δ(t) = −t + 3 − t⁻¹

If two knots have different Alexander polynomials, they are provably different knots — a strong, checkable statement. But the polynomial isn't a perfect invariant: distinct knots can occasionally share the same Alexander polynomial, which is why later invariants (the Jones polynomial from 1984, and the even more powerful HOMFLY and Khovanov homology) were developed to resolve the cases Alexander's polynomial can't.

Why any of this matters beyond puzzles

Knot theory turns out to be the right mathematical language for several very physical problems. Biologists use it to classify how DNA and proteins knot and supercoil, and to understand the topoisomerase enzymes that cut and reseal DNA strands to untangle them during replication. Chemists synthesise genuinely knotted molecules and use knot invariants to distinguish them from their mirror images (chirality matters for how a molecule interacts with other chiral molecules, including most drugs and enzymes). And in physics, knot invariants like the Jones polynomial turned out to be deeply connected to quantum field theory, particularly Chern-Simons theory, linking a 19th-century puzzle about tangled rope to some of the most abstract corners of modern theoretical physics.

Frequently asked questions

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