In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'.^{[citation needed]} Such groups may be described by explicit presentations, as was shown by Emil Artin (1947). For an elementary treatment along these lines, see the article on braid groups. Braid groups are also understood by a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.^{[1]}
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Braids in Higher Dimensions  Numberphile

What is a Knot?  Numberphile

Connections Between Braid Groups, Homotopy Theory, and Low Dimensional Topology  Fred Cohen
Transcription
So let me first tell you what a braid is imagine that you have like two wooden planks and then there are some strings attached In a neat row to the first wooden plank, and then they are all tangled up But then when they arrive to the top they are again in a neat Straight line where they started so like one goes to one two goes to two. Yes So that's called a pure braid I'll only talk about pure braids today It doesn't hugely matter you can also consider ones that that crisscross and end up somewhere else But let me try to draw one that actually goes back to where it started from it goes from Bottom to the top in between you have some tangled mess So when you have some big tangled mess, what's your what's the first question that comes to mind? How would you untangle it? How would you untangle it or can you even untangle it so sometimes you have a big tangled mess And it's entirely possible to untangle it What does it mean to untangle it it means to make it into this braid. This is like the non braid So how do you untangle something? So imagine that the strands are say made of rubber? They are stretchy you can move them around as much as you want You can stretch as much as you want, but you cannot cut them once you cut them the game is ruined So the question is if you see a tangled mess like this Can you make it into a an untangle? This has some practical? Applications for example if you made this video a year ago, I had long hair and when you braid long hair, oh Maybe I can show you on just this little bit so the standard thing you do is you divide it into Three parts this will make me look rather silly And then you do this braid one goes over the middle one and then the other goes over but became the Middle one and you just continue doing this until You reach the end of your hair and then you put an elastic on it. Now it's not two wooden planks But at the top. It's attached to your head and at the bottom It's fastened in an elastic so the top and bottom are kind of fixed the same as in the picture that I just showed you and The good thing about that braid is that it cannot be untangled Because if it could be untangled then as you move your head around it probably would become untangled and the one that I just drew Probably cannot be untangled for example if you look at just the first and the last string you can see that those are Like this, they are they are kind of wrapped around each other in a way that you can't imagine untangling, so Why is it that some braids cannot be undone when you move the strands around move the strings around? What's the problem that you might run into it's that two strands catch on each other? And you can't move them past each other without breaking them So that's the main reason that some braids cannot be undone if the strands would just go through each other magically It ghosts then any braid could be undone. This is in Threedimensional space that we all know very well because we live in it I'm claiming that if you think of braids in four dimensional space then they become completely Uninteresting the main reason is that if you have two strands in four dimensional space they can basically go through each other like ghosts So why is that let me try to give you an analogy first? So there's a little ant that lives in the plane So this ant is twodimensional all it ever knew was this Twodimensional plane that it lives in and this twodimensional plane has a wall the wall is one dimensional it's just a line in the plane, so this little ant Can't go through the wall and will never know What's on the other side of the wall it just hits a red line It just hits a red line, and it can't go through it. What happens if One day a miracle happens and all of a sudden the world of this ant becomes Threedimensional so all of a sudden there's height that day The ant could just go up to the line and say wait But this line has no height at all so I can just jump over it and discover the other side So that's how if you add a dimension things that were uncrossable before are suddenly Crossable because you can use the extra dimension to cross them [Brady] is that assuming that when we kick up to the fourth dimension Our hair doesn't take on an extra dimension with it?[/Brady] exactly yes, exactly, so This wall that the ant that that has kept the ant on one side of its plane For its entire life when you added an extra dimension the wall didn't suddenly become infinitely high in twodimensional So that's a very good point So let me try to show you why this ant picture is relevant to the braids and the fourth dimension? So you have a braid in three dimensions? And you can't untangle it Because somewhere you have two strands that you want to move past each other and you can so this is one strand and Behind it. There's another strand and you Want to move it past what you can't because it gets stuck on this strand. That's in front of it Yeah, so what you want to do is to move this But it's not working Imagine the plane in which this front strand lives and this Strand behind that you're trying to move past Will only intersect that plane in one point right so that point will basically be the ant So if you're just thinking of the picture that you can see in this one plane Then you have the ant which is where the strand behind? Intersects the plane and you have this other strand which lives in the plane And that's the wall and then when I try to move this strand behind and get stuck on this strand That's basically the ant running up against the wall that becomes the hitching point of the two strands that becomes the hitting point But if I add an extra dimension now this is hard to imagine because I mean the the braid is already in threedimensional space so then I have an extra fourth dimension you can imagine that extra fourth Dimension by Imagining that this playing all of a sudden becomes a threedimensional space, okay? so if I add an extra dimension to the whole picture Then there will be an extra dimension added to this plane So in this new space the bottom braid can simply jump over the other strand in the extra dimension So not in the three dimensions that it lived in before, but when you add the extra dimension it can jump over [Brady}Could you still braid your hair if you were a four dimensional being would your hair have a new property that's still made it braidable?[/Brady] Well it depends, so if I stayed the same, but I was placed in four dimensions then no For the reason that I just told you because each strand of my hair is basically one Dimensional [Brady] your DNA would probably all fall apart as well.[/Brady] but if when I was placed in four dimensions I myself became four dimensional and my strands of hair became twodimensional then I could so I Said braids in four dimensions aren't interesting and now I will show you why they are The key is that it's not enough to Braid strings I just showed you that if you're braiding strings that can always be undone in the fourth dimension So I have to show you what to braid instead You can braid something besides strings to show you that first. I like to show you a different way To see just regular braids in three dimensions and since we were talking about ants So you have some ants and I'm making a movie in which the ants start out in a neat straight line and then they crawl around and Eventually they get back to the neat straight line I will film them doing this and then there will be a whole bunch of frames I could imagine just stacking the frames In the first frame I have the four ants in a straight line and then in the second frame They will have moved slightly in the third frame They will have moved again and then so on and so forth until I get to the last frame in which they are again in a straight line and in between if I trace all the Ways that they had moved what we got there is a braid okay? so if I make a movie of ants crawling on a plane Then that's the same as having a braid in 3D just by stacking all the movie frames on top of each other So what does my previous statement mean in this context so my previous statement? Is that braids in four dimensions are uninteresting? What would be braids in four dimensions? Well, it would be like flies flying around in Threedimensional space, and then I make a movie of the flies flying around and I stack them on top of each other so the fourth dimension is the time and the frames of the movie each frame is a Threedimensional thing because they are flying in space, so what I get is braided strings in four dimensions So that is still uninteresting to a mathematician for the same reason why it's uninteresting to you Because it wouldn't hold your hair together. It can always be undone So there's no very interesting analysis that can be done the only point. I want to get out of it Is that you can think of this Uninteresting object as a movie about flies and now I can tell you what the interesting object is instead of making a movie about flies You could make a movie about other things moving around in threedimensional space The one that I actually think about in my research is rings flying around in threedimensional space So if you have some rings they start out in a straight line, and then they can fly around in threedimensional space and Eventually get back to the same position if I make a movie about this and take all these threedimensional movies frames and stack them on top of each other in four dimensions I get something like a braid so your braiding tubes. As opposed to rings, tubes are twodimensional So you're braiding twodimensional things in? fourDimensional space so wire rings more interesting than flies I have some slightly mangled rings here. So if they just fly around not interacting ever they are exactly like flies and they are boring, but Since they are rings. They have a hole inside something they can do is fly right through each other and flying through each other that's Fundamentally different from just being flies, and that's what makes these braided tubes something interesting as opposed to Just braided strings which in four dimensions are uninteresting so they are allowed to Expand and and get smaller and bigger as they need to further purpose of flying through each other as they fly through each other they create the analogous kind of Tangled nests kind of things that you can move past each other right within a braid yeah It's a Tube going through another Tube which would be impossible in three dimensions for a tube to go through another Tube without intersecting it but in four dimensions It's possible, and it catches the same way that strings would catch on each other in three dimensions Can I only go completely inside each other or can they overlap like? No they can't intersect so they are still you know there you can think of them as Physical objects physical object they are not phantom rings the material doesn't go through each other It's just the rings flying through the hole inside the other ring This you can generalize to any dimensions, so in three dimensions you can braid Onedimensional string with a onedimensional string one dimensions with one dimensions one plus one equals two which is three minus one in four dimensions, you can braid a onedimensional with a twodimensional, so that's the fly when you take the Trajectory in time that is the ring when you take the trajectory in time. That's three and three equals four minus one you can also Braid A twodimensional with a twodimensional which is four which is bigger than four minus one and similarly in five dimensions, you couldn't braid a onedimensional with a onedimensional or a onedimensional with a twodimensional but you could you could do two with two or three with three as long as dimensions of your two strings add up to at least the dimension of the Containing space minus one then you can break them together Well in my research, I mostly think about one in two dimensional things Braided or tangled in fourdimensional space or the simplest thing of onedimensional things braided or tangled in threedimensional space and the relationships between them So my favorite way of visualizing. It is is to think of it in terms of movies in three dimensions and That is actually a little bit restrictive because Visualize the fourth Dimension is time and time only moves forward
Contents
Braids as fundamental groups
To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold X of dimension at least 2. The symmetric product of n copies of X means the quotient of X^{n}, the nfold Cartesian product of X by the permutation action of the symmetric group on n strands operating on the indices of coordinates. That is, an ordered ntuple is in the same orbit as any other that is a reordered version of it.
A path in the nfold symmetric product is the abstract way of discussing n points of X, considered as an unordered ntuple, independently tracing out n strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace Y of the symmetric product, of orbits of ntuples of distinct points. That is, we remove all the subspaces of X^{n} defined by conditions x_{i} = x_{j}. This is invariant under the symmetric group, and Y is the quotient by the symmetric group of the nonexcluded ntuples. Under the dimension condition Y will be connected.
With this definition, then, we can call the braid group of X with n strings the fundamental group of Y (for any choice of base point – this is welldefined up to isomorphism). The case where X is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher homotopy groups of Y are trivial.
Closed braids
When X is the plane, the braid can be closed, i.e., corresponding ends can be connected in pairs, to form a link, i.e., a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, depending on the permutation of strands determined by the link. A theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the "closure" of a braid. Compare with string links.
Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. Markov (1935) describes two moves on braid diagrams that yield equivalence in the corresponding closed braids. A singlemove version of Markov's theorem, was published by Lambropoulou & Rourke (1997).
Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid.
The Markov theorem gives necessary and sufficient conditions under which the closures of two braids are equivalent links.^{[2]}
Braid index
The "braid index" is the least number of strings needed to make a closed braid representation of a link. It is equal to the least number of Seifert circles in any projection of a knot.^{[3]} Additionally, the "braid length" is the longest dimension of a braid.^{[4]}
Applications
Braid theory has recently been applied to fluid mechanics, specifically to the field of chaotic mixing in fluid flows. The braiding of (2 + 1)dimensional spacetime trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almostinvariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen–Thurston classification.^{[5]}
See also
 Braid group
 Braided monoidal category
 Change ringing software – how software uses braid theory to model bellringing patterns
 Knot theory
References
 ^ Artin, E. (1947). "Theory of Braids". Annals of Mathematics. 48 (1): 101–126. doi:10.2307/1969218.
 ^ J. S. Birman, Knots, links, and mapping class groups, Annals of Math Study, no. 82, Princeton University Press (1974).
 ^ Weisstein, Eric W. (August 2014). "Braid Index". MathWorld – A Wolfram Web Resource. Retrieved 20140806.
 ^ Weisstein, Eric W. (August 2014). "Length". MathWorld – A Wolfram Web Resource. Retrieved 20140806.
 ^ Boyland, Aref & Stremler (2000); Gouillart, Thiffeault & Finn (2006); Stremler et al. (2011).
Notes
 Birman, Joan S. (1974), Braids, links, and mapping class groups, Annals of Mathematics Studies, 82, Princeton, N.J.: Princeton University Press, ISBN 9780691081496, MR 0375281.
 Boyland, Philip L.; Aref, Hassan; Stremler, Mark A. (2000), "Topological fluid mechanics of stirring" (PDF), Journal of Fluid Mechanics, 403: 277–304, Bibcode:2000JFM...403..277B, doi:10.1017/S0022112099007107, MR 1742169, archived from the original (PDF) on 20110726.
 Fox, R.; Neuwirth, L. (1962), "The braid groups", Mathematica Scandinavica, 10: 119–126, MR 0150755.
 Gouillart, Emmanuelle; Thiffeault, JeanLuc; Finn, Matthew D. (2006), "Topological mixing with ghost rods", Physical Review E, 73 (3): 036311, arXiv:nlin/0510075 , Bibcode:2006PhRvE..73c6311G, doi:10.1103/PhysRevE.73.036311, MR 2231368.
 Lambropoulou, Sofia; Rourke, Colin P. (1997), "Markov's theorem in 3manifolds", Topology and its Applications, 78 (1–2): 95–122, doi:10.1016/S01668641(96)001514, MR 1465027.
 Markov, Andrey (1935), "Über die freie Äquivalenz der geschlossenen Zöpfe", Recueil Mathématique De La Société Mathématique De Moscou (in German and Russian), 1: 73–78.
 Stremler, Mark A.; Ross, Shane D.; Grover, Piyush; Kumar, Pankaj (2011), "Topological chaos and periodic braiding of almostcyclic sets", Physical Review Letters, 106 (11): 114101, Bibcode:2011PhRvL.106k4101S, doi:10.1103/PhysRevLett.106.114101.
External links
 "Braids  the movie" A movie in computer graphics to explain some of braid theory (group presentation, word problem, closed braids and links, braids as motions of points in the plane).
 WINNER of Science magazine 2017 Dance Your PhD: Representations of the Braid Groups. Nancy Scherich.
 Behind the Math of Dance Your PhD, Part 1: The Braid Groups. Nancy Scherich. Explanation of braid groups as used in the movie.