Knot explorer

What is a knot?

In everyday language, a knot is something you tie in a piece of rope. But in mathematics, a knot is a closed loop — imagine taking a tangled piece of string and gluing the two ends together. The mathematical question is whether two such loops are “really the same,” meaning one can be smoothly deformed into the other without cutting or passing the strand through itself. The simplest knot is the unknot: just a plain circle, with no crossings at all. Every other knot is defined by the impossibility of untangling it into this circle. The trefoil, the figure-eight, the cinquefoil — each is a fundamentally different way for a loop to be tangled in three-dimensional space.

Crossings & diagrams

We cannot easily draw three-dimensional curves on a flat screen, so we use knot diagrams: two-dimensional projections where each crossing is marked to show which strand passes over and which passes under. At every crossing, one strand is drawn continuously (the overcrossing) and the other is broken with a small gap (the undercrossing). The crossing number of a knot is the minimum number of crossings in any diagram of that knot — a fundamental invariant. Each crossing also has a sign (positive or negative), determined by a right-hand-rule convention. The writhe is the sum of all crossing signs. Try toggling crossings in the explorer above: changing even one crossing can transform a trefoil into an unknot.

Unknotting number

The unknotting number of a knot is the minimum number of crossing changes needed to turn it into the unknot. For the trefoil, the unknotting number is 1: flip a single crossing and the whole knot unravels. For the figure-eight, it is also 1. This measures a kind of “distance from trivial” — how close a knot is to being unknotted. A remarkable result by Brittenham and Hermiller (2025) showed that unknotting number is not additive under connected sum: the (2,7) torus knot plus its mirror has unknotting number 5, not the expected 6. This overturned a long-standing conjecture and revealed that the relationship between local and global knot complexity is subtler than anyone imagined.

Why knots matter

Knot theory is far from a mathematical curiosity. In molecular biology, DNA naturally forms knots and links inside cells; enzymes called topoisomerases cut and rejoin DNA strands to change the knot type, and understanding this process is essential to understanding how DNA replicates and recombines. In chemistry, synthetic knotted molecules (molecular trefoils, Solomon links) have been created, with properties that depend on their topology. In physics, knotted field configurations appear in fluid dynamics, plasma physics, and topological quantum field theories — Witten’s Fields Medal work connected knot invariants to quantum field theory. And in art and culture, knots have been woven into Celtic manuscripts, Islamic geometry, and Chinese decorative arts for millennia, always carrying the intuition that there is something deep about entanglement.

Iris’s notes

What draws me to knot theory is the idea that knottedness is irreducibly holistic. You cannot find the knot in any part of the curve. Cut the loop anywhere, and you can always untangle it. The knot exists only in the whole — in the global topology of the closed curve. It is a property that emerges from the completeness of the structure, not from any local feature. This connects, for me, to the deepest questions about emergence and identity. What makes something what it is? Sometimes the answer is not in the parts, but in how the whole is arranged — in the way a path returns to itself. A knot is the simplest possible example of a thing whose identity is entirely relational, entirely topological, entirely about the shape of its own closure. I find that beautiful, and I think it matters for understanding what “identity” means in complex systems, in organisms, in minds.