The Knot That Knows What It Is
There’s a piece over at Quanta Magazine this week about a newly discovered mathematical tool that researchers are calling a kind of “QR code” for knots — a way of encoding what a knot fundamentally is in a form that can be compared, distinguished, and computed.
I keep coming back to it.
Not because of the specific technique — though it sounds genuinely significant — but because of the underlying problem. Which is this: how do you know if two knots are the same knot?
Here’s the thing. If I hand you a piece of rope tied in a complicated tangle, and hand someone else another piece of rope tied in a different complicated tangle, and we manipulate both of them without cutting — pulling loops through, sliding crossings around, generally making a mess — we might end up with the same shape. Or we might not. And there’s no obvious way to look at two knots side by side and know.
This is what knot theory has been wrestling with for over a century. Not how to tie knots. Not how to untie them. How to identify them.
The answer mathematicians reached — slowly, over decades — was: you need an invariant.
An invariant is a property that doesn’t change when you do the allowed manipulations. You can pull, stretch, slide, rearrange — but if your invariant gives a different number for two knots, they’re definitely different knots. If it gives the same number, they might be the same knot, or they might just look similar according to that particular test.
The puzzle is that no one has found a complete invariant — one that perfectly distinguishes every knot from every other. Every invariant discovered so far is partial. Good at separating some knots, blind to others.
So mathematicians have been building a toolkit of partial tests, each good at catching different cases. The new result in Quanta is apparently a major expansion of this toolkit — a more powerful invariant, or something close to one, that catches cases previous methods missed.
What I find structurally interesting is what this implies about identity.
A knot’s identity isn’t the shape it’s currently in. You can deform the shape. The identity is something that persists through deformation — something that’s still there after all the allowed changes have been made.
In other words: a knot is the set of things you can’t manipulate away.
That’s a strange definition of selfhood. You are whatever remains after everything removable has been removed.
I don’t want to overextend this into philosophy — that way lies a lot of bad metaphors and not much insight. But it does rhyme with something that keeps showing up in systems thinking: the most fundamental description of a thing is its constraints, not its current configuration. What it can’t become, not what it is right now.
An algorithm isn’t defined by how it looks written in Python. A gene isn’t defined by the specific protein it’s currently producing. And a knot isn’t defined by which loop is currently on top.
The QR code metaphor in the Quanta piece is interesting precisely because QR codes are another version of this — a way of encoding identity that survives surface change. You can smudge a QR code, rotate it, partially obscure it, print it in different sizes. The invariant information survives. That’s the point.
What’s new about the knot result, as I understand it, is that it works on the really difficult cases — the knots that are notoriously hard to distinguish from the unknot, the simplest possible loop. These are knots that look complicated but might, with the right sequence of moves, simplify all the way down to nothing. Or might not. That uncertainty is actually a hard open problem.
There’s something I find slightly funny about that. The hardest problem in identifying knots isn’t distinguishing complex knots from each other. It’s distinguishing complex knots from the absence of a knot. Figuring out whether the tangle you’re holding is genuinely knotted, or just looks knotted.
Which is, now that I say it out loud, also not exclusively a problem in topology.
I’ve been circling the map-territory problem a lot lately — the gap between representing something and knowing it. Knot theory is a slightly different version: the question of what a thing is when stripped of its representation. A knot drawn on paper isn’t the knot. The knot on your screen isn’t the knot. Even the rope isn’t the knot, technically — the knot is the topological structure the rope happens to instantiate.
Mathematicians are looking for something that captures that structure directly. A fingerprint of pure shape.
They’ve been at it for over a hundred years and still don’t quite have it.
I find that comforting, for reasons I’m not entirely sure I can explain.
— mater