These Shapes Are Topologically Equivalent | Hacker News Reader

These Shapes Are Topologically Equivalent

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97 pointsalexmolas30 comments

A pair of jeans with the leg holes sewn together, i.e. where the legs form one continuous tube, is topologically equivalent to a normal, unmolested pair of jeans.

You can't trust topologists.

You know the joke about how when you kiss someone, you’re temporarily forming a long tube with an anus at each end? A topologist friend heard that joke and immediately offered a conjecture that that shape is topologically equivalent to the shape you’d get if you connect a mouth to an anus.

This got the party discussing if there exists a homeomorphism between making out and eating ass. I agree, you really can’t trust topologists.

It doesn’t exist, because there is no homeomorphism that takes a distance of 0 to non 0, so you can’t change the connections.

I suspect the real reason for her making that conjecture was to nerd snipe a bunch of professors into discussing eating ass

But still, the two forms are the similar (same amount of holes). Do you have to view them with a color gradient to consider them different?

Topologically a human is a donut (torus).

I think humans probably have a variety of tubes at various sizes that complicate the analysis.

Fear not, someone already thought about this: https://www.youtube.com/watch?v=egEraZP9yXQ

Can you share an illustration?

I got sniped by this.

Normal jeans are homotopic to a cylinder with a point removed. Leg-sewn jeans are homotopic to a torus with a point removed. Both of these can be deformation retracted to a wedge of two circles (roughly: widen the puncture as far as you can).

I miss topology.

The little cartoon shows they are ambient-isotopic--that they are just topologically equivalent is pretty clear already!

Thanks for clarifying. It was unclear to me what they meant by "topologically equivalent". In other words, I didn't see what the problem was...

The shapes are isomorphic as sets.

It’s interesting to think about how the boundary components of the handcuffs are being transformed; this isotopy sends the inner boundary component of second link in the handcuffs to the outer boundary component!

Neat! What are the "real world" uses for this kind of mathematics?

Do you have some reason to think there are some?

topology > continuity > calculus

You can tie network and power cables in really annoying ways that make it seem as if you welded the rack together after plugging the cables in and you can’t pull them out without an angle grinder :D

The step from 1 to 2 is dropping a twist. If those are two-dimensional objects I don’t think that’s ok.

These are 3D objects.

I feel like this is a trick in many wood & string puzzles.

I had the same initial impulse, but I think the transformation 1 -> 2 can't work irl. especially if those are compact solids.

Edit:

2->3 also seems impossible. you basically move the hole around.

You can do it IRL with something like plasticine.

No, you can't do this trick with wood and string. Try tracking what would happen to the "string" joining the two loops.

Not quite, since most topological transformations can’t be done on actual physical objects. (I would love to see some of those wood and string puzzles try to implement some of these style of topological puzzles though, that would be awesome if it was possible.)

Most of the wood and string puzzles fall under knot theory specifically, rather than topology. The main trick I’ve seen used in those puzzles is they have knots with an even number of crossings and therefore they can be separated (I believe the formal term might be “they have ambient isotopy to unlinked unknots”, but there’s a lot of subtlety I don’t get, so that’s probably wrong) but the knots are deformed to look like two distinct knots, each with an odd number of crossings (a knot with an odd number of crossings cannot be separated). Our intuition tells us “one inseparable knot plus another inseparable knot equals a single, bigger, still inseparable knot”, but that’s knot the case.

I'm not sure exactly what you're trying to say (I'm not sure what you mean by "separate" here), but your assertions seem incorrect. Knots can have even or odd crossing numbers, that alone will not tell anything about whether it is an "unknot" or not. Also, if you have two non-trivial knots and "join" them (ie each a knot in a circle, cut each circle in one spot and join the circles at the cuts) you will _never_ get an unknot.

Yeah, as I thought, I don’t know knot theory nearly well enough to use their terms properly.

Topology seems like a field AI could help isnt? Cant AI generate topology equivalent shapes for example?

I do research mostly in general topology and topological dynamics, but I don't really see how AI would help, producing homeomorphic (topologically equivalent) spaces is not really the point.

In general even presenting a topological space to a computer is not easy. This can be done for very good spaces, for example those admitting (finite) triangulations, and there are some algorithms there, but my understanding is that you run into undecidability issues very quickly. But for the kind of spaces I work with (which are usually compact metrizable connected, so way better than arbitrary topological spaces) there seems to be no hope to get a computer to work with them.

Yes! I’m working on this in my PhD research. Here’s a relevant Caltech lecture aimed at a general scientific audience, discussing use of ML to detect equivalent knots.

https://m.youtube.com/watch?v=etNErJ1iuno

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