Untangling the beautiful math of KNOTS

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Suppose you have two tangled knots. How can you tell whether they are the same knot or different knots? Welcome to Knot Topology. A knot is a nice embedding of a circle into three dimensional space, up to "ambient isotropy" where you can wiggle it around but not cut it. We can project this onto a plane to create a knot diagram. Then, two knot diagrams represent the same knot if and only if there is a sequence of Reidermeister moves between them. We next explore various knot invariants like the simpler tricolorability and then expand to the Alexander Polynomial and ultimately compute this polynomial for the trefoil knot.
The software I used to make all the knots is called knotplot.com/
0:00 Mathematical Knots
0:33 Knot Diagrams
1:48 Reidermeister Moves
4:24 The equivalence problem
6:05 The idea of Knot Invariants
6:27 Tricolorability
9:12 Alexander Polynomial
14:37 Future direction in Knot Topology
16:01 Brilliant.org/TreforBazett
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Опубликовано:

1 авг 2024

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Комментарии : 104

@kanishkachakraborty 2 года назад

Love the topology t-shirt, and incredibly interesting video as always. Thank you!

@DrTrefor 2 года назад

Thank you!

@kanishkachakraborty 2 года назад

My understanding of how the 1st and 3rd Reidermeister moves preserve tricolourability: 1st: A section of the knot is crossing itself, so only 1 colour is used - tricolourability allows a crossing having only a single colour. 3rd: If the two initial crossings satisfied tricolourability, the move preserves it, because sliding a section of the knot can only shift the position of crossings without modifying the nature of the crossings.

@greghearn7428 2 года назад

I absolutely love knot math. Great to see such a nice breakdown of it.

@DrTrefor 2 года назад

Thank you! It really is so cool

@brazni 2 года назад

I was tired of being practical all the time so I got into knot theory

@3moirai 2 года назад

Great introduction to knot theory!

@DrTrefor 2 года назад

Thank you!

@billycox475 2 года назад

I'm here because I was just trying to figure out how to get an extension cord untangled

@DrTrefor 2 года назад

lol did I help you?

@billycox475 2 года назад

@@DrTrefor it's in an elegant unknotted coil now. Plus, I learned something. So time well spent. Great channel!

@shortsismakingmybrainrot 2 года назад

Omg this is cool, I absolutely love your channel, thanks so much for helping me get through my uni math courses.

@NonTwinBrothers 2 года назад

Best video explaining the concept of know theory I've ever seen. Well not that I'd be able to understand it way back then but you know

@jcreazy 2 года назад

I came here because I wanted to know how knots work. Now I'm more confused. Fascinating video. Thanks for making it.

@aashsyed1277 2 года назад

The title is so good so how good can the video get ? 10⁹ times better.

@abrahammekonnen 2 года назад

Great video

@DrTrefor 2 года назад

thank you!

@BleachWizz Год назад

Nice video, definetly going into my references for my article! Also I have something to add: 3:18 - but just Reidemeister moves in sequence is not enough to directly simplify any diagram. By that I mean it's not always identifiable that you can perform a Reidemeister move to remove a crossing. I'm not sure though if moves that INCREASE the number of crossings are necessary, I THINK that only movements that keeps the number of crossings the same would have a chance to allow some undoing, but I could be wrong and I'd love to know.

@seslocrit9365 2 года назад

Could you do a video of DNA and Knots? Also, (on an unrelated topic) could you do a video on non-linear dynamics?

@DrTrefor 2 года назад

oooh, i wonder what connections there are to DNA!

@pseudolullus 2 года назад

@@DrTrefor many! Topology, winding number and twisting are crucial in DNA biology. It's crucial for DNA replication, bacterial plasmids and even cancer treatments which target aptly named topoisomerase proteins

@mahmoudalbahar1641 2 года назад

Many thanks for your great videos. And I suggest making video about non-integer base of numeration.

@DrTrefor 2 года назад

You are most welcome! That would be a fun topic for sure:)

@lgl_137noname6 2 года назад

4:20 to 4:25 amazingly, Google subtitle AI managed to not make a spelling mistake in the script. 6:09 I spoke too soon . 7:50 Knot invariant is definitely throwing it a curve ball.

@DrTrefor 2 года назад

haha that's kinda crazy how good the AI is these days, especially given how "not" and "knot" sound similar and this is a very isoteric topic!

@482man 2 года назад

I once tried to make preztels with different knots, but it was too hard so I ended up with a plate of tri-knots lol

@muzammilaziz9979 2 года назад

It's Reidemeister, with no r in between.

@DrTrefor 2 года назад

Lol oops:D

@simonsays_999 4 месяца назад

i love knots :3

@crsmith6226 Год назад

Me listening to this: of course they’re different knots, they’re in different places duh.

@maxp3141 2 года назад

Wow, this video is just knots… apologies, I couldn’t resist it. :)

@DrTrefor 2 года назад

@crytp0crux Год назад

Just discovered the secret to Picasso art. They sort of look like knots. Doesn't the second one in 1:16 look like a Picasso drawing of Mr. Potato Head? That gives us a "Mr. Reidermeister Potato Head" by Picasso.

@Nebukanezzer 7 месяцев назад

Small error. The height of the power tower is 10^(1000000*n), not 10^1000000^n.

@angusritchie1956 Год назад

how did you choose which line was yellow or purple for the Alexander Polynomial?

@oriole8789 2 года назад

Thanks for your videos! I'd like to bring your attention to the lower volume of some of your videos. If you right click on this video and select "stats for nerds" you can see that the content loudness is -14.6dB, where it should really be closer to 0dB. Since this is a log scale, the gap is substantial. Some of your videos are definitely mixed at significantly quieter volumes than others. In practice it just means that people would have to turn up the volume quite a bit, but that might make their next video play loud in a jarring way. Depending on the software that you use for editing, it may be possible to include a "compressor" filter in the audio chain, which can be used to normalize the audio levels and reduce the audio's dynamic range which will make it easier to hear on devices like phones, laptops etc, in addition to getting the output to be closer to 0dB. There are lots of tutorials on how compressors work (it's standard fair for radio and TV broadcast). Thanks!! -Nick

@DrTrefor 2 года назад

Thanks for letting me know, I'll do some more research:)

@ethandavis7310 2 года назад

In the case where you take the un-knot and perform R-move 1, you'll end up with one crossing and 3 regions. Based on the diagram you showed and the information you gave, it seems that there are two possible values you could enter into the crossing 1-region 3 element of the matrix. How does this work?

@DrTrefor 2 года назад

You can choose either, and then we have to prove (not done in the video) that the knot invariant really is invariant based on this choice you mention as well as the others I talked about, that is giving the same polynomial up to multiplication by t^s

@PeterPrevos 2 года назад

It seems that the unknot only have one region and 0 crossings (or x crossings and x+1 regions). Love to see a video about drawing knot projections in tikz

@DrTrefor 2 года назад

Indeed! I don't actually use tikz for this, i use knotplot mentioned in the description

@PeterPrevos 2 года назад

@@DrTrefor I toyed with knotplot, but find it hard to get to nice knot projections you see in the literature. I am writing about knot theory in magic tricks.

@gonzogil123 2 года назад

Do you have a video on the specific algorithmic procecess that generate knots. Their If then, i.v, d.v. generative functions? Is that available, or, did Disney purchased the rights not to be able to teach it (people have told me that). I know that it would be the functions for the geometry of things you may imagine etc. Like whatever Platonic solid in your head with whatever features etc.

@OeshenNix Год назад

I had my volume on low and thought this was Zach star

@DrTrefor Год назад

We are going to merge into the same person imo

@rostkgb 2 года назад

I Knew there was math behind them😅

@motherisape 2 года назад

I don't understand how definition of topology relates to torus and cup

@DrTrefor 2 года назад

You might like more intro to topology video here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-IpkzNeS8G20.html

@motherisape 2 года назад

@@DrTrefor thanks

@angelmendez-rivera351 2 года назад

The torus and the cup are examples of sets within a topological space, especifically the topological space we associate with 3-dimensional Euclidean space. As sets within this topological space, they are equivalent. They can be effectively treated as if they were the same set. This is because these sets are homeomorphic.

@shutriMedia Год назад

Does this "three colorability" has something to do with three fundamental color charges of Quantum Chromodynamics ?

@SuperDeadparrot Год назад

If you shake a knot to unravel it, it will always unravel in the same direction, even if you try to twist in the opposite direction it will reverse itself.

@Jack_Callcott_AU 2 года назад

The Reidemeister moves have inverses 2) and 3) are their own inverses and we can create an inverse to 1) then we can say two knots K_1 and K_2 are related by a relation R such that K_1 R K_2 iff there is a sequence of Reidemeister moves from K_1 to K_2. R is reflexive , symmetric and transitive and is therefore an equivalence relation on the set of knots which partitions the set into different knot-types. Am I not correct? BTW, thanks for the video. I have never seen this before.

@DrTrefor 2 года назад

Indeed!

@Jack_Callcott_AU 2 года назад

@@DrTrefor

@hdheuejhzbsnnaj 2 года назад

Fantastic, but what about a full course! 😉

@DrTrefor 2 года назад

That would be amazing! I've never done a full grad level course on RU-vid before, but if I did knot topology would be a great topic for it

@hdheuejhzbsnnaj 2 года назад

@@DrTrefor absolutely. Most of the grad level math on RU-vid is pretty dry and uninspired in it's presentation.

@waltermelo5538 2 года назад

Greetings! This was a very interesting and inspirating video, please can you recomend us some bibiography to study knot topology? I did my master in algebraic topology and I know things like homotopy, homology and cohomology. Thank you so much for your content.

@DrTrefor 2 года назад

Here is the notes for one course on knot topology: www.math.toronto.edu/~drorbn/classes/20-1350-KnotTheory/

@waltermelo5538 2 года назад

@@DrTrefor Thank you so much!

@anhthiensaigon 2 года назад

I have an intuition that when we already have a 2D projection of a knot, and start walking from an arbitrary point on the string. Whenever we walk over a cross, we take note whether the section of the string that crosses our path lies over or under our path (can be saved as a chain of 1s and 0s). Then out of that binary chain, we can recreate exactly 1 knot which is identical to the original knot, and we can also perform some calculations over it. Did mathematicians already consider this possibility? If yes, and if you know any proofs that this method wouldn't work, can you show us? Thanks :)

@DrTrefor 2 года назад

How do you keep track of WHERE you cross, is it between two other crossing for instance?

@PeterPrevos 2 года назад

This is a bit like the Dowker-Thistletwaithe notation

@KurdaHussein 2 года назад

how did u know zerez 1 in region V?

@parth123ify 2 года назад

Have people used neural nets to distinguish between knots? What's the performance?

@DrTrefor 2 года назад

That is a really interesting idea, I haven't seen such research but my GUESS here is that it is going to run into computability problems because a lot of the core problems in knot theory come down to the challenge of having insanely large number of computations for even very small knots. maybe neural nets can sidestep some of that mess in exchange for a small error rate or something of this nature.

@StaticBlaster 2 года назад

I believe this is used a lot in superstring theory (M-theory).

@robheusd Год назад

Knots do not exist in dimensions higher then 3 (or lower)

@StaticBlaster Год назад

@@robheusd sure. I'm just going by the website that was online 5 years ago. They took it down. I'm not sure why but it was showing what math topics you need to know in order to be a string theorist.

@suhana.a.a7949 4 месяца назад

Please provide the reference textbook sir

@continnum_radhe-radhe 2 года назад

This totally new topic for me ...it is strange

@GeoffryGifari 2 года назад

is there a pattern on how many knots thete are for a given number of crossing?

@DrTrefor 2 года назад

Not that I am aware of, but this function f(n) for the number of unique knots with n crossing definitely grows extremely fast. As I mentioned f(23) is over a 100 billion.

@GeoffryGifari 2 года назад

@@DrTrefor i'm thinking there's gotta be *some* pattern, this being math haha

@andrewharrison8436 Год назад

You hid what D&D players call a plot hook in there: "... can be calculated in polynomial time". So people have discovered invariants that can't be calculated in polynomial time?

@hala2um960 2 года назад

what is about matric space???

@continnum_radhe-radhe 2 года назад

🔥❤️🙏

@Eduardo-cr8ri 2 года назад

Is this video for one of your math classes too?

@DrTrefor 2 года назад

Nope, just a cool advanced topic I wanted to share with RU-vid

@jakubb4784 Год назад

Is there a perfect knot invariant?

@DrTrefor Год назад

Sadly no computable “complete invariant”

@crpfx302 2 года назад

💗💗💗💗🧡🧡

@henrik3141 2 года назад

Related to this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-aMxcAaR0oHU.html

@GeoffryGifari 2 года назад

hmmm maybe if we have several knot invariants, we can uniquely identify every knot

@DrTrefor 2 года назад

The real goal is a “complete invariant” which means it goes both directions, two knots are the same if and only if the invariant is the same. Sadly we don’t have such a computable complete invariant for knot theory

@abrahammekonnen 2 года назад

13:26 could you define what a well-defined polynomial is?

@godfreypigott 2 года назад

So you're looking for a well-defined well-defined polynomial?

@abrahammekonnen 2 года назад

@@godfreypigott No I meant what does well-defined mean. Either I didn't understand the definition he gave(in which case could someone please restate it) or he was using a circular definition(which is what it seemed like to me).

@godfreypigott 2 года назад

@@abrahammekonnen Ughhh ... when someone takes a joke literally ....

@abrahammekonnen 2 года назад

@@godfreypigott oh lol sorry

@DrTrefor 2 года назад

A definition is "well-defined", loosely, if it results in the same thing regardless of choices of input. So in our case given a knot there are many choices of knot diagram, many choices for labeling it, ambient isotropy, many choices for which columns of the matrix to eliminate etc. So you have to prove that for all those choices, it gives the same polynomial.

@theleviathan3902 Год назад

This video is about "not topology"? dang it

@missoss 2 года назад

Could you please consider backing up your channel on Rumble and/or Odysee?

@Neptoid 2 года назад

The knots are too small! I can't distinguish the crossings

@Philoreason 2 года назад

The sound volume is way too low.. otherwise good stuff!

@Whereareugari 2 года назад

Turn up the volume

@godfreypigott 2 года назад

Knot maths is not maths. Knot!

@ChannelMath Год назад

you didnnt define what 'consecutive regions' are. you just labeled the regions I-V seemingly arbitrarily. Thanks love your channel!

@amaanabbasi280 2 года назад

Voice is soo low sir

@weetabixharry Год назад

Is this joke funny? No, it's knot.

@swordofstrife1174 2 года назад

Knot theory is honestly my absolute favorite field of mathematics, learning about it is what got me interested in math beyond school Here's an awesome book on the subject: www.math.cuhk.edu.hk/course_builder/1920/math4900e/Adams--The%20Knot%20Book.pdf

@DrTrefor 2 года назад

Thanks for sharing!