This article contains a list of mathematical knots and links. See also list of knots, list of geometric topology topics.
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Transcription
Hi. I want to talk about understanding, and the nature of understanding, and what the essence of understanding is, because understanding is something we aim for, everyone. We want to understand things. My claim is that understanding has to do with the ability to change your perspective. If you don't have that, you don't have understanding. So that is my claim. And I want to focus on mathematics. Many of us think of mathematics as addition, subtraction, multiplication, division, fractions, percent, geometry, algebra  all that stuff. But actually, I want to talk about the essence of mathematics as well. And my claim is that mathematics has to do with patterns. Behind me, you see a beautiful pattern, and this pattern actually emerges just from drawing circles in a very particular way. So my daytoday definition of mathematics that I use every day is the following: First of all, it's about finding patterns. And by "pattern," I mean a connection, a structure, some regularity, some rules that govern what we see. Second of all, I think it is about representing these patterns with a language. We make up language if we don't have it, and in mathematics, this is essential. It's also about making assumptions and playing around with these assumptions and just seeing what happens. We're going to do that very soon. And finally, it's about doing cool stuff. Mathematics enables us to do so many things. So let's have a look at these patterns. If you want to tie a tie knot, there are patterns. Tie knots have names. And you can also do the mathematics of tie knots. This is a leftout, rightin, centerout and tie. This is a leftin, rightout, leftin, centerout and tie. This is a language we made up for the patterns of tie knots, and a halfWindsor is all that. This is a mathematics book about tying shoelaces at the university level, because there are patterns in shoelaces. You can do it in so many different ways. We can analyze it. We can make up languages for it. And representations are all over mathematics. This is Leibniz's notation from 1675. He invented a language for patterns in nature. When we throw something up in the air, it falls down. Why? We're not sure, but we can represent this with mathematics in a pattern. This is also a pattern. This is also an invented language. Can you guess for what? It is actually a notation system for dancing, for tap dancing. That enables him as a choreographer to do cool stuff, to do new things, because he has represented it. I want you to think about how amazing representing something actually is. Here it says the word "mathematics." But actually, they're just dots, right? So how in the world can these dots represent the word? Well, they do. They represent the word "mathematics," and these symbols also represent that word and this we can listen to. It sounds like this. (Beeps) Somehow these sounds represent the word and the concept. How does this happen? There's something amazing going on about representing stuff. So I want to talk about that magic that happens when we actually represent something. Here you see just lines with different widths. They stand for numbers for a particular book. And I can actually recommend this book, it's a very nice book. (Laughter) Just trust me. OK, so let's just do an experiment, just to play around with some straight lines. This is a straight line. Let's make another one. So every time we move, we move one down and one across, and we draw a new straight line, right? We do this over and over and over, and we look for patterns. So this pattern emerges, and it's a rather nice pattern. It looks like a curve, right? Just from drawing simple, straight lines. Now I can change my perspective a little bit. I can rotate it. Have a look at the curve. What does it look like? Is it a part of a circle? It's actually not a part of a circle. So I have to continue my investigation and look for the true pattern. Perhaps if I copy it and make some art? Well, no. Perhaps I should extend the lines like this, and look for the pattern there. Let's make more lines. We do this. And then let's zoom out and change our perspective again. Then we can actually see that what started out as just straight lines is actually a curve called a parabola. This is represented by a simple equation, and it's a beautiful pattern. So this is the stuff that we do. We find patterns, and we represent them. And I think this is a nice daytoday definition. But today I want to go a little bit deeper, and think about what the nature of this is. What makes it possible? There's one thing that's a little bit deeper, and that has to do with the ability to change your perspective. And I claim that when you change your perspective, and if you take another point of view, you learn something new about what you are watching or looking at or hearing. And I think this is a really important thing that we do all the time. So let's just look at this simple equation, x + x = 2 • x. This is a very nice pattern, and it's true, because 5 + 5 = 2 • 5, etc. We've seen this over and over, and we represent it like this. But think about it: this is an equation. It says that something is equal to something else, and that's two different perspectives. One perspective is, it's a sum. It's something you plus together. On the other hand, it's a multiplication, and those are two different perspectives. And I would go as far as to say that every equation is like this, every mathematical equation where you use that equality sign is actually a metaphor. It's an analogy between two things. You're just viewing something and taking two different points of view, and you're expressing that in a language. Have a look at this equation. This is one of the most beautiful equations. It simply says that, well, two things, they're both 1. This thing on the lefthand side is 1, and the other one is. And that, I think, is one of the essential parts of mathematics  you take different points of view. So let's just play around. Let's take a number. We know fourthirds. We know what fourthirds is. It's 1.333, but we have to have those three dots, otherwise it's not exactly fourthirds. But this is only in base 10. You know, the number system, we use 10 digits. If we change that around and only use two digits, that's called the binary system. It's written like this. So we're now talking about the number. The number is fourthirds. We can write it like this, and we can change the base, change the number of digits, and we can write it differently. So these are all representations of the same number. We can even write it simply, like 1.3 or 1.6. It all depends on how many digits you have. Or perhaps we just simplify and write it like this. I like this one, because this says four divided by three. And this number expresses a relation between two numbers. You have four on the one hand and three on the other. And you can visualize this in many ways. What I'm doing now is viewing that number from different perspectives. I'm playing around. I'm playing around with how we view something, and I'm doing it very deliberately. We can take a grid. If it's four across and three up, this line equals five, always. It has to be like this. This is a beautiful pattern. Four and three and five. And this rectangle, which is 4 x 3, you've seen a lot of times. This is your average computer screen. 800 x 600 or 1,600 x 1,200 is a television or a computer screen. So these are all nice representations, but I want to go a little bit further and just play more with this number. Here you see two circles. I'm going to rotate them like this. Observe the upperleft one. It goes a little bit faster, right? You can see this. It actually goes exactly fourthirds as fast. That means that when it goes around four times, the other one goes around three times. Now let's make two lines, and draw this dot where the lines meet. We get this dot dancing around. (Laughter) And this dot comes from that number. Right? Now we should trace it. Let's trace it and see what happens. This is what mathematics is all about. It's about seeing what happens. And this emerges from fourthirds. I like to say that this is the image of fourthirds. It's much nicer  (Cheers) Thank you! (Applause) This is not new. This has been known for a long time, but  (Laughter) But this is fourthirds. Let's do another experiment. Let's now take a sound, this sound: (Beep) This is a perfect A, 440Hz. Let's multiply it by two. We get this sound. (Beep) When we play them together, it sounds like this. This is an octave, right? We can do this game. We can play a sound, play the same A. We can multiply it by threehalves. (Beep) This is what we call a perfect fifth. (Beep) They sound really nice together. Let's multiply this sound by fourthirds. (Beep) What happens? You get this sound. (Beep) This is the perfect fourth. If the first one is an A, this is a D. They sound like this together. (Beeps) This is the sound of fourthirds. What I'm doing now, I'm changing my perspective. I'm just viewing a number from another perspective. I can even do this with rhythms, right? I can take a rhythm and play three beats at one time (Drumbeats) in a period of time, and I can play another sound four times in that same space. (Clanking sounds) Sounds kind of boring, but listen to them together. (Drumbeats and clanking sounds) (Laughter) Hey! So. (Laughter) I can even make a little hihat. (Drumbeats and cymbals) Can you hear this? So, this is the sound of fourthirds. Again, this is as a rhythm. (Drumbeats and cowbell) And I can keep doing this and play games with this number. Fourthirds is a really great number. I love fourthirds! (Laughter) Truly  it's an undervalued number. So if you take a sphere and look at the volume of the sphere, it's actually fourthirds of some particular cylinder. So fourthirds is in the sphere. It's the volume of the sphere. OK, so why am I doing all this? Well, I want to talk about what it means to understand something and what we mean by understanding something. That's my aim here. And my claim is that you understand something if you have the ability to view it from different perspectives. Let's look at this letter. It's a beautiful R, right? How do you know that? Well, as a matter of fact, you've seen a bunch of R's, and you've generalized and abstracted all of these and found a pattern. So you know that this is an R. So what I'm aiming for here is saying something about how understanding and changing your perspective are linked. And I'm a teacher and a lecturer, and I can actually use this to teach something, because when I give someone else another story, a metaphor, an analogy, if I tell a story from a different point of view, I enable understanding. I make understanding possible, because you have to generalize over everything you see and hear, and if I give you another perspective, that will become easier for you. Let's do a simple example again. This is four and three. This is four triangles. So this is also fourthirds, in a way. Let's just join them together. Now we're going to play a game; we're going to fold it up into a threedimensional structure. I love this. This is a square pyramid. And let's just take two of them and put them together. So this is what is called an octahedron. It's one of the five platonic solids. Now we can quite literally change our perspective, because we can rotate it around all of the axes and view it from different perspectives. And I can change the axis, and then I can view it from another point of view, but it's the same thing, but it looks a little different. I can do it even one more time. Every time I do this, something else appears, so I'm actually learning more about the object when I change my perspective. I can use this as a tool for creating understanding. I can take two of these and put them together like this and see what happens. And it looks a little bit like the octahedron. Have a look at it if I spin it around like this. What happens? Well, if you take two of these, join them together and spin it around, there's your octahedron again, a beautiful structure. If you lay it out flat on the floor, this is the octahedron. This is the graph structure of an octahedron. And I can continue doing this. You can draw three great circles around the octahedron, and you rotate around, so actually three great circles is related to the octahedron. And if I take a bicycle pump and just pump it up, you can see that this is also a little bit like the octahedron. Do you see what I'm doing here? I am changing the perspective every time. So let's now take a step back  and that's actually a metaphor, stepping back  and have a look at what we're doing. I'm playing around with metaphors. I'm playing around with perspectives and analogies. I'm telling one story in different ways. I'm telling stories. I'm making a narrative; I'm making several narratives. And I think all of these things make understanding possible. I think this actually is the essence of understanding something. I truly believe this. So this thing about changing your perspective  it's absolutely fundamental for humans. Let's play around with the Earth. Let's zoom into the ocean, have a look at the ocean. We can do this with anything. We can take the ocean and view it up close. We can look at the waves. We can go to the beach. We can view the ocean from another perspective. Every time we do this, we learn a little bit more about the ocean. If we go to the shore, we can kind of smell it, right? We can hear the sound of the waves. We can feel salt on our tongues. So all of these are different perspectives. And this is the best one. We can go into the water. We can see the water from the inside. And you know what? This is absolutely essential in mathematics and computer science. If you're able to view a structure from the inside, then you really learn something about it. That's somehow the essence of something. So when we do this, and we've taken this journey into the ocean, we use our imagination. And I think this is one level deeper, and it's actually a requirement for changing your perspective. We can do a little game. You can imagine that you're sitting there. You can imagine that you're up here, and that you're sitting here. You can view yourselves from the outside. That's really a strange thing. You're changing your perspective. You're using your imagination, and you're viewing yourself from the outside. That requires imagination. Mathematics and computer science are the most imaginative art forms ever. And this thing about changing perspectives should sound a little bit familiar to you, because we do it every day. And then it's called empathy. When I view the world from your perspective, I have empathy with you. If I really, truly understand what the world looks like from your perspective, I am empathetic. That requires imagination. And that is how we obtain understanding. And this is all over mathematics and this is all over computer science, and there's a really deep connection between empathy and these sciences. So my conclusion is the following: understanding something really deeply has to do with the ability to change your perspective. So my advice to you is: try to change your perspective. You can study mathematics. It's a wonderful way to train your brain. Changing your perspective makes your mind more flexible. It makes you open to new things, and it makes you able to understand things. And to use yet another metaphor: have a mind like water. That's nice. Thank you. (Applause)
Contents
Knots
Prime knots
 0₁ knot/Unknot  a simple unknotted closed loop
 3₁ knot/Trefoil knot  (2,3)torus knot, the two loose ends of a common overhand knot joined together
 4₁ knot/Figureeight knot (mathematics)  a prime knot with a crossing number four
 5₁ knot/Cinquefoil knot, (5,2)torus knot, Solomon's seal knot, pentafoil knot  a prime knot with crossing number five which can be arranged as a pentagram
 5₂ knot/Threetwist knot  the twist knot with threehalf twists
 6₁ knot/Stevedore knot (mathematics)  a prime knot with crossing number six, it can also be described as a twist knot with four twists
 6₂ knot  a prime knot with crossing number six
 6₃ knot  a prime knot with crossing number six
 7₁ knot, septafoil knot, (7,2)torus knot  a prime knot with crossing number seven, which can be arranged as a {7,2} star polygon
 7₄ knot, "endless knot"
 8_{18} knot, "carrick mat"
 10₁₆₁/Perko pair
 12n242/(−2,3,7) pretzel knot
 (p, q)torus knot  a special kind of knot that lies on the surface of an unknotted torus in R^{3}
Composite
 Square knot (mathematics)  a composite knot obtained by taking the connected sum of a trefoil knot with its reflection
 Granny knot (mathematics)  a composite knot obtained by taking the connected sum of two identical trefoil knots
Links
 02
1 link/Unlink  equivalent under ambient isotopy to finitely many disjoint circles in the plane  22
1 link/Hopf link  the simplest nontrivial link with more than one component; it consists of two circles linked together exactly once (L2a1)  42
1 link/Solomon's knot (a two component "link" rather than a one component "knot")  a traditional decorative motif used since ancient times (L4a1)  52
1 link/Whitehead link  two projections of the unknot: one circular loop and one figure eightshaped loop intertwined such that they are inseparable and neither loses its form (L5a1)  63
2 link/Borromean rings  three topological circles which are linked and form a Brunnian link (L6a4)  L10a140 link  presumably the simplest nonBorromean Brunnian link
 Brunnian link  a nontrivial link that becomes trivial if any component is removed
 Pretzel link  a Montesinos link with integer tangles