@johncarlosbaez@mathstodon.xyz
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johncarlosbaez

@johncarlosbaez@mathstodon.xyz

I'm a mathematical physicist who likes explaining stuff. Sometimes I work at the Topos Institute. Check out my blog! I'm also a member of the n-Category Café, a group blog on math with an emphasis on category theory. I also have a YouTube channel, full of talks about math, physics and the future.

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johncarlosbaez, (edited ) to random
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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.

See the wild knot here? There's just one point where the stuff it's made of 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. This is hard to draw. I'd really like to see a good try.

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.

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https://www.youtube.com/watch?v=o7U3yvMF8Sw

johncarlosbaez, (edited )
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For the hard-core math lovers out there, let me say a bit more about this paper:

• Vadim Kulikov, A non-classification result for wild knots, https://arxiv.org/abs/1504.02714.

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:

Kulkov 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 ∼ 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 reasonably nice sort of topological space, and a Borel relation is a reasonably nice sort of relation on such a space. I don't want to stun you with the definitions - they're easy to look up on Wikipedia.

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. There's a theory of when you can reduce one such classification problem to another. This is what Kulikov used to state and prove his result.

At this point you start noticing that the word 'logic' is hiding inside the word 'topology'. Probably not a coincidence.

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johncarlosbaez,
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To see even more on this - including all the definitions I just left out - read my blog article:

• Wild knots are wildly difficult to classify, https://golem.ph.utexas.edu/category/2024/05/wild_knots_are_wildly_difficul.html

I've got a bunch of questions for category theorists and logicians at the end - especially ones who like the connections between logic and topology. (You know who you are.)

Here's the best picture I've seen so far about a knot that's wild at every point, called the 'Bing sling'. It's not a very good picture (since some bits of the knot just fizzle out in mid-air), but maybe you can guess how it should work.

I want better pictures of everywhere wild knots!

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