Untangling the beautiful math of KNOTS

[@kanishkachakraborty]

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

[@DrTrefor]

Thank you!

[@greghearn7428]

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

[@DrTrefor]

Thank you! It really is so cool

[@HelPfeffer]

0:02 I genuingly thought you were going to say "Are they the same not, or knot?"

[@kanishkachakraborty]

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.

[@NonTwinBrothers]

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

[@3moirai]

Great introduction to knot theory!

[@DrTrefor]

Thank you!

[@aashsyed1277]

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

[@shortsismakingmybrainrot]

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

[@abrahammekonnen]

Great video

[@DrTrefor]

thank you!

[@BleachWizz]

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.

[@jcreazy]

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

[@mahmoudalbahar1641]

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

[@DrTrefor]

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

[@brazni]

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

[@seslocrit9365]

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

[@DrTrefor]

oooh, i wonder what connections there are to DNA!

[@pseudolullus]

@@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

[@lgl_137noname6]

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]

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!

[@oriole8789]

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]

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

[@ethandavis7310]

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]

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

[@angusritchie1956]

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

[@Jack_Callcott_AU]

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]

Indeed!

[@Jack_Callcott_AU]

@@DrTrefor

[@muzammilaziz9979]

It's Reidemeister, with no r in between.

[@DrTrefor]

Lol oops:D

[@482man]

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

[@simonsays_999]

i love knots :3

[@abrahammekonnen]

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

[@godfreypigott]

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

[@abrahammekonnen]

@@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]

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

[@abrahammekonnen]

@@godfreypigott oh lol sorry

[@DrTrefor]

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.

[@crytp0crux]

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.

[@PeterPrevos]

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]

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

[@PeterPrevos]

@@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.

[@hdheuejhzbsnnaj]

Fantastic, but what about a full course! 😉

[@DrTrefor]

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

[@hdheuejhzbsnnaj]

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

[@GeoffryGifari]

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

[@DrTrefor]

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]

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

[@shutriMedia]

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

[@maxp3141]

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

[@DrTrefor]

:D

[@KurdaHussein]

how did u know zerez 1 in region V?

[@hala2um960]

what is about matric space???

[@anhthiensaigon]

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]

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

[@PeterPrevos]

This is a bit like the Dowker-Thistletwaithe notation

[@SuperDeadparrot]

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.

[@suhana.a.a7949]

Please provide the reference textbook sir

[@parth123ify]

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

[@DrTrefor]

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.

[@waltermelo5538]

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]

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

[@waltermelo5538]

@@DrTrefor Thank you so much!

[@jakubb4784]

Is there a perfect knot invariant?

[@DrTrefor]

Sadly no computable “complete invariant”

[@billycox475]

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

[@DrTrefor]

lol did I help you?

[@billycox475]

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

[@OeshenNix]

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

[@DrTrefor]

We are going to merge into the same person imo

[@StaticBlaster]

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

[@robheusd]

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

[@StaticBlaster]

@@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.

[@Eduardo-cr8ri]

Is this video for one of your math classes too?

[@DrTrefor]

Nope, just a cool advanced topic I wanted to share with KZbin

[@continnum_radhe-radhe]

🔥❤️🙏

[@gonzogil123]

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.

[@Nebukanezzer]

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

[@crpfx302]

💗💗💗💗🧡🧡

[@crsmith6226]

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

[@continnum_radhe-radhe]

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

[@fernandocolima7561]

Our body is a toroid because of our digective tube

[@motherisape]

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

[@DrTrefor]

You might like more intro to topology video here: kzbin.info/www/bejne/f6HOq4GbiJ16aJI

[@motherisape]

@@DrTrefor thanks

[@angelmendez-rivera351]

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.

[@henrik3141]

Related to this video: kzbin.info/www/bejne/l37blHSXh5Wifrc

[@rostkgb]

I Knew there was math behind them😅

[@andrewharrison8436]

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?

[@missoss]

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

[@Philoreason]

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

[@Whereareugari]

Turn up the volume

[@GeoffryGifari]

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

[@DrTrefor]

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

[@Neptoid]

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

[@amaanabbasi280]

Voice is soo low sir

[@godfreypigott]

Knot maths is not maths. Knot!

[@theleviathan3902]

This video is about "not topology"? dang it

[@ChannelMath]

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

[@AdemonGamer09]

Who else came here after finishing chapter 7 of reverse 1999

[@weetabixharry]

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

[@swordofstrife1174]

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]

Thanks for sharing!