How The Most Useless Branch of Math Could Save Your Life

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My earphones discover all 352,152,252 knots half a second after entering my pocket.

Veritasium's graphic designer would have become a physicist by now. It must take a lot of understanding on their part to learn the concept first and then visualize how to demonstrate it. It is just mind-blowing.

Although I wouldn't in a hundred years choose to study knots, it never fails to inspire me how people take up these causes, pushing esoteric fields forward for nothing more than a pure love of the game until eventually their discoveries can be tied to other fields to make important scientific advances. Science really is humanities most collaborative sport.

"Not knot"
"Who's there?"
The *Unknot*

There is a certain demographic that I hope never finds this extended tribute to knots

I'm in awe at someone having the patience and skill to visually tie thousands of knots and distinguish duplicates. Thanks for another great story wonderfully told.

I don’t think I’ve ever watched a video that I completely understood while, at the same time, I had no idea what you were talking about. It’s like a miracle. Fascinating subject!

A buddy of mine is studying knots in projective spaces, called like “knots in the shadow world”; he explained that you can’t necessarily get a well defined projection onto the plane for them (over/under crossings could be the same so you don’t really get a “drawing” of the knot in way that works well).

As a Ph.D. student in algebraic topology, I am glad to see that different areas of topology are finally getting the attention they deserve.

Fun fact: making sure you're doing a square knot instead of a granny knot is also very important in surgery. They are taught to alternate the way they secure off the suture.

i love that I never know in which rabbit hole I am going to fall into when I watch your videos but it`s always incredibly fascinating! Thank you for your valuable content!

Hello internet, welcome to Knot theory

Math is so incredible. People just study a phenomenon and it reveals a language that describes the world in ways we didn't know of before. Wonder what fields of maths exist that we have yet to study

When it comes to avoiding knots in headphones, my easy way that I have used for years is to simply not allow the endpoints into the storage case. If it is a zippered pouch, just leave the earbuds and the plug hanging outside the zipper. You can just grab the entire length of cable and shove it in the zipper pouch as a massive wad if you want, just leave the ends outside the zipper, and you will have no knots. Same story if you put them in your pocket. Just leave the ends sticking out of the pocket and you get no knots.

I've started to lace my shoes like this ~2 years ago when I realized its symetry and aesthetism, but never paid attention to its tightness. Thank you for this great video, as always !

This is the most complicated video I've ever watched in this channel. I didn't Understand much but after seeing that our everyday knot can be so interesting to great scientific minds I get excited. 😯

Derek has been killing it with the math videos lately

My great aunt (Mary Gertrude haseman) was one of the founders of knot theory (after tait) (incidently her brother got his ph.d under Hilbert). She worked out a subset of 12 crossings in the early 1900w. After ph.d she became a housewife and knots were ignored until around the 80s mostly

Many of your videos i watch and enjoy your explanations yet only begin to grasp the concepts. Some I’m still clueless yet i love to know how much i don’t know about the world. Thank you for them all.

17:34
I lost track on the 5-colorable with this image showing only 4 colors

In southern India, everyday women create knot diagrams called 'Kolam' in their front porches . The more complicated the knotting the more skilled the person drawing it. This video is when I realized that many such Kolams are actually super complicated, challenging to create, un-knots! So cool to know there is a whole sub branch of math around this!
Unknowingly it sounds like these women have been practicing Reidemeister moves on a daily basis in attempts to create ever more complicated knot diagrams!
And instead of the p-colourability, these Kolams care more about the areas between the loops which are denoted by simple points. The points are actually laid out first and the knot emerges as a line diagram around them .

I’m so glad that veritasium is making more and more math videos.

@31:36 I stiffened knowing the mistake you were about to broadcast as good practice... never do this with any type of cable, but especially cable that contains un-braided stranded wire Ie. Earbuds
3 destructive actions take place when you twist stranded wire onto itself, or even wrapped around a center bobbin. Ie. An extension cord spooled around your hand and elbow.
1. The individual wires that make up stranded wire are equal in length (for many reasons) but as you twist the cable, each individual wire inside the protective insulator is rotating longitudinally, and when you un-twist your cable the inner wires do not completely return to their factory position, so over time they begin to bunch around each other tighter and tighter. since the wires closest to the outside are effectively getting shorter faster, having a farther distance to move with each rotation, they will begin to knot inside the cable first, snapping away from the ends while the inner wire continue to become stiffer, and exerting force on the cable ends in the opposite direction as the outer wires. After a while you will notice that even when you spool out the cable it begins to look wavy as if it had been curled. Since there is less wire making contact with each snapped strand, the resistance slow rises at each end of the cable dropping the voltage... In the case of extension cords you will notice that this makes the plug act like a heating element. With earbuds you will notice that over time the volume decreases, until shortly before one side stops working they only operate at 25%-50% of original volume. You can actually measure it, check output when new, then twist and untwist them 180 times @6 months worth of simulated use, and measure the difference.
2. Friction, you ever had to cut a wire or piece of metal in half, and had only your hands, you folded it several times at a certain point... well a twist is essentially a rotational fold and each crossing point is a fold location since copper is a soft metal, each time you force one side of wire to cover a longer distance than the other side what happens is the two opposing forces pinch the atoms together at that spot forcing them to rub against each other with much more force than they would see during normal use. To figure out how much farther a side has to pinch and stretch to accomplish a twist, take the diameter of the wire then find the circumference for 180 degrees, now using that distance as your radius, double it to find the diameter of the twisted wire, now find the circumference using that diameter for 180 degrees and that is the distance of travel between the shortest side, and longest side.
3. With stranded wires, the insulation is usually not as tough as cable with less strands of thicker gauge, simply for the same reason we use stranded wire in applications where we want the cable to remain easily pliable. So the insulation is easier to damage inside the cable after repeated use, and the insulation be it enamel for earbuds or pliable vinyl in romex, with enough twist, you can find yourself with a cable that is in a permanent state of "dead short" and the big danger being it quickly melts the conductor ending the short, so it gets set aside, and you forget or a coworker uses the cord, plug it in making it live, the rolls the cord out loop by loop and when they get to the place where the damage is, the movement causes their hand to complete the circuit at the damaged location.
I tell people all the time the two best ways two best ways to store any cable is do not twist or loop decide on a convenient length and fold the cord back and forth like an accordion then pinch the centers together and tie with a piece of scrap wire or bread tie depending on size of cable. The second way is essentially the same as the first, but instead of a back and forth stacking of the cable, start a spiral on a flat surface with then roll it around itself so that it has no rotation perpendicular to the length like it does when looping, now continue that rolling, until you have four to five passes on the flat spiral. Now start your next roll on top of the first roll at the same size make the same number of passes you did on the first spiral layer, repeat until you have a roll of cord 4-5 passes wide and about the same number of layers high. you will almost certainly be left with a cable end on each side or both ends meeting at same place. You can now place a tie around the finished coil. I almost always use two so that it retains its shape for storage... It I am unclear you can visualize it as being rolled the same way windings on a transformer are.

This is amazing Im so glad I watched this till the end to see what it could be used for. It’s a lot to digest and I have a certain problem concentrating but I made it. Thank you for posting.

Just to avoid potential confusion for those folks moving to a squar knot for tying their shoes, its actually isnt just clockwise or counter clock wise, it depends on how the first overhand knot you make is tied, whether its the left side going over then under or vice versa. It also depends on if you make the loop on the left or right side. So check to look if it actually appears like a squar knot to confirm.

This video changed my life. I don’t have to double knot my shoes now. I tied them the other way and they didn’t come undone all 12 hours running around at work. Amazing.

The knot in my back after sitting for 8 hours at work has got to be a new discovery

Thanks for what you do. This is insanely informative and well presented.. Very informative and entertaining!!.

My initial reaction was "So what? I wear slip-on shoes.". I then became slightly interested by the mathematics underlying knot theory and ended up floored by the practical applications of this area of study. An excellent video as always.

Veritasium always makes a knot into my brain with his genius explanations, but I did not expect it to be literally about knots.

As always, I learn so much from your videos. Thank you for sharing!

I'm not a matematician with fine expertise on knots... but i know something about storage of audio matherial.
I could clearly hear those hearbud's chord screaming for mercy!
The first thing you want to avoid is twisting cables, that's why it is most commonly stored in circular or "8" shape takeing care to gently twist it between your finger while coiling up so the core is kept straight. To prevent further mess there are chord holders or just use some tape. Confineing in a restricted area as a bag is a good advice anyway.

20:50 In grad school I took a class with Jozef Przytycki on Graph Theory and Knots, and he was also on my oral exam committee. It's so cool to see him pop up in a Veritasium video!
This has to be one of the best, if not THE best, video on knot theory on KRplus. Amazing job as always, Veritasium!

when you realise there is not an impostor at 27:49...

I have no idea how you make these subjects fascinating but it is working flawlessly

Ending the video with this quote was immensely satisfying and tied (!) everything together so elegantly.

As a knitter and never heard of this is so fascinating. As someone who hated math but use basic arithmetic for knitting; but this field of math wants me advanced my math knowledge and thinking.

Oh man, I got excited when Conway showed up. It's fascinating every where and every time he shows up. In this one he just pops in, does something in an afternoon that no one had done before and then we don't hear from him anymore. I'm very grateful for all the footage we have of him.

Your video was literally amazing. Thank you for sharing such incredible knowledge. After watching your video, I got interested in the relationship between the knot theory and materials in chemistry. So I tried to find some more about it, but it was hard. Could you recommend some good resources/researches/websites/articles or whatever related to it? Thank you in advance!

Excellent video as always. I'm pretty much always learning something new in every video you make, thank you for that.
Now about the tying you shoelaces bit......The clock-wise or counter-clock-wise way isn't really the point.
Let me explain.
When tying your shoelaces what you essentially are doing is tying a two knots. But by not pulling any further at the point where the loops are about as long as the remainder of the shoelaces you end up with the laces in a nice looking bow. Continue to pull the loops all the way until the end of the laces are through and you'll have made either a granny knot or a square knot, and are left with long shoe laces again.
Whether you'l have to go clock-wise or not when tying the bow-bit actually depends on the first knot you've tied. Like you explain in the video around 29:15, in essence you wanna ty a square knot because that doesn't loosen up as easy.
When tying a normal square knot you go Left over Right and then Right over Left, or vice versa. As long as you counter the first knot when making the second knot you'll end up with a square knot.
So depending on how you tie the first knot, let's say the base for the bow-part, you need to go clock-wise or counter-clock-wise with the bit that forms the bow-part in order to get a square knot at the end.
If during your day you have to tie your she's again you'll will have made a granny knot the first time which loosens up while you are walking.
Once again thanks for the all you have taught me in all of your videos.

I was very surprised when Derek, a scientist, missed a very important piece of the shoe tying equation. You would only tie your shoes (bows) clockwise if you first tied the base knot counter clockwise, otherwise you are just tying a "lefthanded" granny knot. Clockwise or counter clockwise makes no difference, a proper knot involves one of each. Eg. left over right + right over left = good knot. You can't just finish correctly if you've started wrong in the first place.

Making something as potentially boring as knot-theory interesting is no small feat. You, sir, are simply a truly gifted educator.

Working with paracord while i watch this. Ive been so obsessed with learning new knots lately so this vid hits the spot, and may explain why people like me find them so fascinating.

I really enjoyed this video.
Knots have fascinated me from childhood. I figured out on my own how to tie certain knots that amazed and frustrated others. I especially liked making varieties of slip knots because people would be trying to figure it out, and I would pull and the complex knot would vanish like a magic trick. And I would laugh and evil laugh.
I was a weird kid.

As a contractor and a math enthusiast, I quickly learned that isolating the two free ends of a long extension cord would make any wrapping process into an unknot. Many of us know the braid, which I believe is a type of sailor’s way of keeping a rope from tangling. I also knew IT people who’d use similar strategies for long cat-5 cables. It would be interesting to me to see how tradespeople and sailors had long ago “discovered” these theories by trial and error.
Incidentally, I’ve tied my shoes with a square knot since I was a teenager, and my young kids know the difference between the granny and square. When they try to teach their friends they’re usually met with blank stares. 😂

0:00 Intro
0:24 Knot Theory
1:17 What is a knot?
3:40 Knot equivalence problem
4:32 Other famous knots in history
5:20 Vortex theory of the atom
8:25 On Knots paper (the first seven order of knottiness)
9:52 Reidemeister Moves
10:51 Haken’s Unknot Theorem (and upper bound and crossing number)
13:01 Knot invariant
14:13 Tricolorability
16:37 p-colorability (and polynomials)
21:34 Perko pair
22:32 n-crossings knots
24:21 Molecular knots
28:31 How You Should Knot Your Shoes :)
29:40 Doug Smith & Dorian Raymer experiments
32:00 Knot Theory's Potential
33:15 Outro (and video sponsor Brilliant)

For the shoe tying thing it's about the direction of tension applied to the knot vs the direction the knot tightens or loosens upon

Very interesting! As a casual student of physics in by browsing I often encounter connections between knots and Quantum Field Theory (QFT), which I have previous had a hard time understanding. This video has helped me understand them better. Thanks a lot!

Knot theory, and particularly the Alexander- and Jones polynomials, were my first foray into mathematics research in 2013. So happy to see algebraic topology getting a spotlight on your channel, and the incredible applications!

when I was six and extremely bored, I had a habit of tying hair bands into as many knot combinations as I could think of. I also realized that they behaved differently than if I tied a knot with a normal string, and I've always been curious to find out why. This was super informative and kept my attention the whole way through, great video!

that graph showing how many knots that certain prople discovered kept blowing my mind over and over again when more people were added

I was expecting mention of knotphysics. Knots are used to explain space time physics in manifolds. Fundamental particles (u,d,e,...) could still be considered as knots.
Great video as always. Thanks.

As a budding molecular biologist, I know think that it is essential and indeed impossible to venture into the field without having a thorough grasp of knot theory first. Thanks Veritasium, truly eye-opening.

I’ve been a rock climber for over five years and this was a great video. I’ve spent a lot of time wondering about knot mechanics. and you spent a lot of time explaining knots, and I really appreciate that. I learned a lot.

One of the most interesting videos I've seen in a long time. Frankly, the math goes WAY over my head. I'm not nearly studied enough to understand the polynomial equations, at least. Still, the premise is so intriguing for the fact that it proves with math that there are true answers to a seemingly impossible question.
And, honestly, if there was ever an application for which quantum computers are best suited, this may just be the one. But what would be even more exciting is if they can't solve for very large prime knots. Because the equations which form them could become the basis of future encryption the way large prime numbers are today.

The square knot you showed towards the end is (at least in Germany) used in Firefighting. Its main Use is to connect two ropes together because it pulls itself tighter when pressure is applied

Very informative and entertaining!!

The fix for turning your granny knots into reef knots is really simple. When you start to tie your shoes, pay attention to which lace you place over the other. This is a habitual action and people do it the same way every time. If your habit is to start by placing your left lace over the right, simply reverse that step, go right over left instead. Then just finish tying your knot the same way you always do. Your hopeless granny knot will become a perfect reel knot. It may take a few times of consciously reversing that initial step, but it will quickly become your new habit and your shoes will never come untied again.

Only at 13:10 so idk if they are already get to this, but I’m thinking that if you just label one crossing as 1, the next as 2, so on and so forth; sort of treating it like a sequence of crossings. you can eliminate the number of crossings by going through the three rightamiester wtv processes to reduce each crossing to simply a non overlapping line that connects the two crosses next to it. Once you’ve done that to each crossing you should know how many true crossings you have, and as they’re already all in a sequence, you can just go along the sequence to mark the type of crossing they are to determine wether they are different knots from another.
Edit: example
Triknot - triquetra would have 3 true crossings and you could label them binarily as 101, or 010 depending on if you start at the top of a crossing or the bottom. To simplify maybe always start on the top of a crossing but I haven’t thought about all the difference and if there are any, could be beneficial to consider both startings as possible in any knots to be compared.
if any two knots have the same number of crossings and their binary sequence can be matched up identically, than they are the same knot.
I wonder if a top starting knot sequence could match up with a bottom starting knot sequence but then each knots opposite start couldn’t match to eachother, that certainly couldn’t make sense if all crossings were true but I’m not trying to think about allat

Thank you for the continuation of education!

30:29 Derek masterfully sneaking in an alibi for all those baggies he’s stashing

It's not whether you counter clockwise or clockwise tie the knot. It depends on the first step and whatever way you cross the strings, it must be the opposite in the second step.

You should really make a video on how you make videos. Each time I'm most stunned how easy it is to follow the logic. Chapeau!

I’m actually working in this area of study right now! There’s more to it as well: in 1996 one Dr. Kauffman invented virtual knot theory to generalize knot invariants to knots embedded in surfaces of arbitrary genus, allowing for knots whose gauss codes (the invariants he was studying) could not exist in 3-dimensional Cartesian space. In short, This is an exciting field of mathematics, precisely because it’s so novel.

Tip to get used to tying shoe laces stronger: don't change the complex movement that finishes the tying, but the first simple knot. The effect is the same
Happy side effect of the secure knot is that the loops stay perfectly perpendicular to the shoe so it's also prettier

The research for your videos must be absolutely insane! Not to mention figuring out and understand those scientific papers to really break it down for us

What a rollercoaster of emotions this video has been.

Reminds me of the lesson on DNA supercoiling and gyrase I had in bio class recently

30:06
"Longer agitation time leads to a higher chance of knotting"
Can confirm.

Its honestly incredible how much your videos have improved in recent years

I will save this to watch later. It is late, and I am knot following along very well.

The Tricolouribility move to solve some part of the knot problem was just genuinely genius

OMG it is uncredible that something so "common" like a knot would help us to develop ENORMOUS changes in science and technology like that. That's the reason I love science.

25:28
No, the most complex knot ever created is my bedsheet twisted and tangled up when I need to use it.

I find kelvins early interpretation of atoms as different knots very interesting due to its surface level similarities to string theory.

the square vs granny thing also applies to tying a jacket around the waist. if you do the overhands in the same direction you get a granny knot that works loose. i did this for years and had to keep tightening the knot. but if you do the overhands in opposite directions you get the square/reef knot which won't work loose.

Derek got it wrong in the very beginning. The two ways of tying your shoe don’t have to do with clockwise or counterclockwise. It has to do with if you do the second crossing in the same direction or not. So like if you did the first crossing clockwise, do the loop part counter clockwise to keep the knot balanced and torques low. You can also do the first crossing counter clockwise if you want instead, as long as you do the opposite on the loop part. It will be just as strong.

FOR TYING SHOE LACES:
You don't need to loop the other way for more security. You can just go around twice. Loop around twice. Not double knotting, but looping twice. It has more friction.

A lot of people are here to see the actual shoe knot explanation at 28:40. My addition to this is that, you can still do a counterclockwise motion and end up with a square knot instead of a granny knot, but to get this, your first half-hitch / overhand knot needs to be done the opposite way as how Derek has it in the video. In his half-hitch, the end pointing to the right is going over and then under, but if you invert this, then once you get to the second part of the knot, it is now when you can do a counterclockwise motion and get a square knot.
Knots can be mirrored, so it isn't really the best explanation to tell it as an absolute that "you have to do it clockwise to get a square knot", because that is not necessarily true, but for this, something else in the process of making the knot needs to change to mirror it properly.

Haha I’ve been using that headphone trick since I figured it out in high school and was very happy about it! Cool to know there’s so much intricate theory behind it!

I bet every time knot theorists get together for a conference they'll be sporting some of the most impressive tie knots you'll see.
I'd hope there's a little competition at each conference to identify the most practical, creative and/or aesthetic tie knot

It doesn't matter wether you tie shoe laces clockwise or counter clockwise around the loop. It all depends on the direction of the first trefoil.

Great video! Interdisciplinary studies is where it's at!

This will help me greatly in finding the one tightest knot that will never break.

I’m surprised that knot theory - a branch of math that is “knot” so easy to understand- can be explained so well!
Mad respect

31:24 For every twist you introduce, you also introduce forces that will eventually snap the thin conductor inside. That's why you should not coil an electrical cable, or a braided climbing rope, like you coil a 3- or 4-strand rope. For your headphones, just doubling and redoubling them to a convenient length and then putting a clothespeg on them is just as efficient and gives you more use from them before you break them. 😊

I was enthralled by this presentation.

btw: The trick to tying shoelaces with a secure and proper square knot doesn't require relearning a clock or counterclock bow. The (slippery reef) square knot doesn't care which way the bow is rotated as long as it is the opposite of the initial (marlin hitch) overhand knot. If your shoelaces are sloppy granny knots, continue using your same familiar bow tying technique. Simply tie your initial overhand knot the other way round.
/ sailor

I never knew knots would be so complicated and diverse in use! It is great to learn about their different uses and I am excited to see what the future could bring for knots!

I love how often seemingly trivial intellectual pursuits lead to great discoveries

freaking love this channel. All the incredibly esoteric things i know.

24:57 This here... This RIGHT HERE is the key to the future... To EVERYTHING

The reef knot is one of the easiest knots to untie one handed. Grabbing one end and giving it a sharp jerk turns the other end into a hitch which slides off thestanding end. It's why it's used to reef sails; you can keep to the old adage: one hand for the ship, the other for you.

I seriously love this channel, seems like these videos take a LOT of effort to build but the quality and completeness and depth is unmatched all in one place.

In my university there was a Institute for knot theory which had one professor and one phd student at a time. They had one small room in the main building of the university. When the professor retired the institute was closed.

To tie a square knot, you take the two ends and go right over left and then left over right. Or you go left over right and then right over left. Technically, when you were showing the counter-clockwise vs clockwise generating a granny vs square, it was because you started with left over right on your leg and then did another left over right (counter-clockwise). If you start tying your shoes right over left, you can then do a counter-clockwise (left over right) to generate a square knot. So it is more about how you start the knot... the second part just has to be the opposite.

I really liked how this video shows the benefits of "knowledge for knowledge's sake". That something with no obvious practical application can be worked on for centuries, and suddenly a breakthrough happens that turns the previously "useless" knowledge into something potentially world changing.

Your shoelace knot also depends on the orientation of the first crossing. I start in the opposite way that you do for the simple reason that the "granny knot shoelace" will sit paralell to your shoe while the sailors knot sits perpendicular. Good to know its also stronger :)

I move for a series of trading cards that catalog all of the prime knots as shown in this video. You can get the unknot and trefoil knot cards as Commons, and a 17-crossing knot as a Super-Rare. Trade them at your local comic-book store.