The step from 1 to 2 is dropping a twist. If those are two-dimensional objects I don’t think that’s ok.
These Shapes Are Topologically Equivalent
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Re: These Shapes Are Topologically Equivalent
#12A 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.
Re: These Shapes Are Topologically Equivalent
#13The 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.
Re: These Shapes Are Topologically Equivalent
#14I feel like this is a trick in many wood & string puzzles.
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.
Re: These Shapes Are Topologically Equivalent
#15It’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!
Re: These Shapes Are Topologically Equivalent
#16A 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.
Re: These Shapes Are Topologically Equivalent
#17A 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.
Can you share an illustration?
Re: These Shapes Are Topologically Equivalent
#18Topology seems like a field AI could help isnt? Cant AI generate topology equivalent shapes for example?
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.
Re: These Shapes Are Topologically Equivalent
#19A 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…
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.
Re: These Shapes Are Topologically Equivalent
#20I feel like this is a trick in many wood & string puzzles.
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 kno…
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.