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|dw:1436886176531:dw| Mirror images of each other without removing the black fixed points.
i can tie a knot on a cherry stem with my tongue :)
lol sasha gray, love your work!
rotating it 180 degrees seems to do the job, let me grab my shoelace and give it a try... |dw:1436925191461:dw|
thanks doll. hmu 4 discounts on movies :)
tried on a simple knot on my shoe lace n it worked :) real question is how to prove it mathematically - i will pass on reading up topology!
yeah im still messing with my shoelace
I'll reveal the answer in about 15 minutes unless anyone wants me to give them more time.
The knot theory is optional, so if you don't care just look at the pictures haha. First we take our knot and we tighten up that bottom one like this: |dw:1436928710044:dw| From this perspective the little knot is just part of the string, and we can slide it out of the other knot like this: |dw:1436928775217:dw| Ahhh now notice we have really what's called a connected sum of two prime knots. The trefoil knot and its mirror image. By itself the trefoil knot is chiral, which means it's either left or right handed and you can't deform it into its mirror image. However it's not what's called "fully chiral" because it's "invertible". What's invertible mean to us and this problem? We'll get there in a second, but invertible just means that you can put an orientation on the knot and then change the orientation and get the same knot back, not its mirror image or anything else, you get the same knot, here, I'll show you: |dw:1436928963697:dw| Notice how you can't tell if I reversed the orientation of the arrows on the string or I actually just flipped my left and right sides of the string. Try this yourself. This is what makes it invertible. So now let's look at the original problem, we now have two knots that are mirror images of each other, and we can slide them past each other in either way we like, placing their mirror images on top of where the old one was, like this: |dw:1436929099805:dw| So now this connected sum of knots is deformable to its mirror image, which is to say it's amphichiral! In fact, it's more than that, it's fully amphichiral because it's invertible as well! So here the knot theoretician can stop and say, "aha, I have shown this knot is fully amphichiral which means it's deformable to its mirror image" and that's what we wanted to show, correct? However let's just go ahead and finish deforming it. To do that, we have to go back and remember that the "little knot" we made and pulled out will now become the "big knot" and have the other one become the "little knot and slide inside of it, like this: |dw:1436929354477:dw| Hopefully that was fun to play with :P
Ok I think I ended up lying, the knot being invertible didn't really matter to us, but whatever, deal with it.
medal 4 giving me a headache; not to mention this: |dw:1436930163396:dw|
Also, its not proof unless you provide the homeomorphism :) else it's just a picture.
A picture is proof! I just got off the phone with Euclid he said it was fine. If you can physically do it after seeing some steps, how does that not constitute proof?
I did not invent topology, I only study it.
also I am just messing around :)
Also, every proof in Elements stands with no picture...
outside of the axiom argument....
well that was a long wait for nothing :)
Haha I had nothing to say, I think I was looking at another tab and pressed some keys on accident in the post box. There's nothing wrong with a proof by demonstration. :)
depends on how you define proof. It would not get published in certain publications...
There is a reason they defined the xy plane and topology. You can prove this axiomatically.
A digital representation of your explanation (done in Blender): |dw:1437014757932:dw| Thanks for the question!
This should not be news to you that if you take an upper level math class and your teacher wants a proof, you cant just draw a picture. Euclid did NOT just draw pictures for proof. You can say what ever you want, but showing a homeomorphism by picture is not a proof. I can draw a never ending staircase, that does not mean it exists.
Blender is a powerful \(free\) 3D tool. Can be used to create meshes (like people & objects), sculpting, animation and motion capture -- it also has texturing, audio integration, green screen capabilities and lots more. I used a Bezier path and a Bezier circle to form this string along with using vectors to move it into the various positions you see here. Get the software here - http://www.blender.org/ My favorite instructor - http://gryllus.net/Blender/3D.html (Niel Hirsig) Here is a movie "Sintel" that was made with this tool - https://www.youtube.com/watch?v=eRsGyueVLvQ I used this extremely simple technique to make the string - "Loft beveling along a path" https://vimeo.com/channels/blendervideotutorials/44841825 I just use my laptop for development -- i.e. doesn't require significant hardware.
All I do is write proofs :) Just google it man. Demonstration is NOT proof. "In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments" This is wiki \(\uparrow\) Now you are probably thinking, well how the f do we do that with knots...its called topology and you may now google homeomorphism. Anyone can take a clay cup and deform it (without tearing it) into a doughnut. Proving it can be done takes topology.