May 26, 2024

Wild Knots are Wildly Difficult to Classify

Posted by John Baez

In the real world, the rope in a knot has some nonzero thickness. In math, knots are made of infinitely thin stuff. This allows mathematical knots to be tied in infinitely complicated ways — ways that are impossible for knots with nonzero thickness! These are called ‘wild’ knots.

Check out the wild knot in this video by Henry Segerman. There’s just one point where it needs to have zero thickness. So we say it’s wild at just one point. But some knots are wild at many points.

There are even knots that are wild at every point! To build these you need to recursively put in wildness at more and more places, forever. I would like to see a good picture of such an everywhere wild knot. I haven’t seen one.

Wild knots are extremely hard to classify. This is not just a feeling — it’s a theorem. Vadim Kulikov showed that wild knots are harder to classify than any sort of countable structure that you can describe using first-order classical logic with just countably many symbols!

Very roughly speaking, this means wild knots are so complicated that we can’t classify them using anything we can write down. This makes them very different from ‘tame’ knots: knots that aren’t wild. Yeah, tame knots are hard to classify, but nowhere near that hard.

As I mentioned, he proved wild knots are harder to classify than any sort of countable structure describable using first-order classical logic with countably many symbols. And it’s interesting how he proved this. He proved it by studying the space of all knots.

So he used logic to prove a topology problem is hard — but he also used topology to study logic!

More precisely:

Kulikov studied the topological space of all knots, which are topological embeddings $K$ of the circle in the 3-sphere. He also studied the equivalence relation on knots saying $K \sim K'$ if there’s a homeomorphism of the 3-sphere mapping $K$ to $K'$.

This is an example of a ‘Borel relation on a Polish space’. A Polish space is a topological space $X$ homeomorphic to a complete separable metric space. A Borel relation is a relation $R \subseteq X \times X$ that’s a Borel set. For more about the definitions, click the links.

A lot of classification problems can be thought of this way: you give a Polish space of things you’re trying to classify, and an equivalence relation saying when two count as ‘the same’, which is a Borel relation. We then say a Borel relation $R \subseteq X \times X$ is Borel reducible to a Borel relation $S \subseteq Y \times Y$ if there’s a Borel function $f: X \to Y$ such that

$R(x,x') \iff S(f(x), f(x'))$ for all $x, x' \in X$

In this situation people say the classification problem $(X,R)$ can be Borel reduced to the classification problem $(Y,S)$.

This is what Kulikov used to state and prove his result. As far as I can tell, he showed:

1) Equivalence of countable models of any first-order theory with countably many symbols can be Borel reduced to equivalence of (possibly wild) knots.

2) Equivalence of knots is not Borel reducible to the equivalence of countable models of any first-order theory with countably many symbols.

At this point you start noticing that the word ‘logical’ is hiding inside the word ‘topological’.

It’s interesting to see how Kulikov proved his result — his paper is so well-written that you can follow the overall logic without sinking into the weeds of detail.

H. Friedman and L. Stanley showed that the space of countable models of any first-order theory with countably many symbols is Borel reducible to a single one of these, coming from the theory of linear orders.

This is pretty surprising to me: I wouldn’t have guessed that classifying countable linear orders was maximally difficult in this sense.

But thanks to this, to prove 1) Kulikov just needs to show:

1$^\prime$) Equivalence of countable linear orders can be Borel reduced to equivalence of (possibly wild) knots.

For 2), he uses a general result due to Hjorth. Suppose that a Polish group $G$ (a group in the category of Polish spaces) acts on a Polish space $X$ in a ‘turbulent’ way — some sort of highly chaotic way, defined in Kulikov’s paper. Then the Borel relation

$x \sim x' \iff \exists g \in G \; g x = x'$

is not Borel reducible to equivalence of countable models of any first-order theory with countably many symbols!

So Kulikov just needs to show

2$^\prime$) The group of homeomorphisms of the 3-sphere acts in a turbulent way on the space of topological embeddings of the circle in the 3-sphere.

Connections to category theory

How do, or could, categorical logicians think about questions like this?

For example, what do categorical logicians think about the problem of classifying countable linear orders? Is there a sense, similar to the one sketched above, in which it’s maximally hard among some class of problems? Or does dropping the axiom of choice dramatically change its status?

Also: what do they think about the topology of the space $X$ of countable models of a first-order theory (which Kulikov says is homeomorphic to the Cantor set)?

I imagine $X$ is the space of objects of a topological groupoid, where the isomorphisms are the usual isomorphisms of models. But Kulikov merely equips $X$ with the relation of “isomorphicness”. That’s how the makes it into a Polish space with a Borel equivalence relation.

Similarly, since we have the Polish group of homeomorphisms of $S^3$ acting on the Polish space of embeddings $K : S^1 \to S^3$, the action groupoid of this groupoid should be a ‘Polish groupoid’. But Kulikov instead treats it as a Polish space with a Borel equivalence relation.

Posted at May 26, 2024 12:23 PM UTC

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Re: Wild Knots are Wildly Difficult to Classify

Does this division of knots into tame and wild extend to higher dimensions - aka knotted surfaces - and does Kulikov’s thm extend to this context?

Posted by: Mozibur Rahman Ullah on May 28, 2024 1:01 PM | Permalink | Reply to this

Re: Wild Knots are Wildly Difficult to Classify

We can generalize the definition of wild knot to higher dimensions. Kulnikov’s theorem took a lot of work, and he proved it only in 2015, so I doubt anyone has already generalized it to higher dimensions. But when someone tries, they should post a comment here!

Andrew Ranicki says that “Zeeman [316] and Stallings [274] proved that embeddings $k : S^n \to S^m$ with codimension $m − n \ge 3$ are unknotted in the piecewise linear and topological categories. Thus topological knotting only starts in codimension 2.” But linking is another story, and his book mostly about locally flat (i.e., non-wild) knots.

Posted by: John Baez on May 28, 2024 3:30 PM | Permalink | Reply to this

Re: Wild Knots are Wildly Difficult to Classify

Thanks for that.

Does that mean we can get knotted surfaces in codimension 1? As that’s not possible with ordinary knots - we can’t knot a curve in the plane.

Posted by: Mozibur Rahman Ullah on June 5, 2024 7:10 AM | Permalink | Reply to this

Re: Wild Knots are Wildly Difficult to Classify

DIRC? that a theorem of Lebesgue asserts that a continuous increasing map of the unit interval to itself is differentiable almost everywhere… If so, could this provide a supply of exotic (Brownian/Wienerian)unknot

(… and perhaps a new subfield of very prickly-feeling topology)?

Posted by: jack morava on June 4, 2024 5:43 PM | Permalink | Reply to this

Re: Wild Knots are Wildly Difficult to Classify

The video by Segerman has pictures of wild knots that rather look like fractals. I’ve not come across fractal knots before and so I googled it. I came across this site:

https://www.mathartfun.com/FractalKnots/index.html

(I would have linked to it but I’m not sure how to do this on this site - is this the href html tag?)

which shows how we can use ordinary knots as a repeating motif. For example the trefoil knot gives something that rather looks like the Sierspinski gasket - except its a wild knot. The more complex knots give knots that rather look like celtic ornamentation. I’m guessing these knots will have a countable number of wild points.

Posted by: Mozibur Rahman Ullah on June 5, 2024 7:23 AM | Permalink | Reply to this

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