What is Topology, and what's the big deal?

Discussion in 'Physics & Math' started by arfa brane, Nov 21, 2014.

  1. rpenner Fully Wired Valued Senior Member

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    The Fourier transform works on any signal, periodic or not, but periodic signals are nice because the Fourier transform has non-zero values only at the frequencies that fit evenly into the period. Thus integration can be replaced with an infinite sum during reconstruction.

    I don't think anyone was saying the Fourier transform would exactly reproduce a signal with discontinuities. We know that it doesn't because the Fourier transform of a signal which is everywhere zero but at one point where it has some finite value, is zero.

    But for a piecewise continuous function, at every continuous point the reconstruction from Fourier components sums to the value of the function at that point and at the discontinuities, it sums to the average of the limits on either side. The Gibbs phenomena is about how this convergence is slow when the derivatives aren't also continuous.

    http://en.wikipedia.org/wiki/Gibbs_phenomenon#Explanation

    I'm late, but here's a periodic function with discontinuities in the derivative:
    \(f(x) = \frac{8}{\pi^2} \sum_{n=1,3,5,...}^{\infty} \frac{(-1)^{\frac{n-1}{2}}}{n^2} \sin \frac{n \pi x}{L} = \frac{2}{\pi} \sin^{-1} \sin \frac{\pi x}{L}\)
    For x = L/2 we have:
    \(f(L/2) = \frac{8}{\pi^2} \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)^2} \sin \frac{(2n + 1) \pi}{2} = \frac{8}{\pi^2} \sum_{n=0}^{\infty} \frac{1}{(2n + 1)^2} = 1\) so there is no particular problem at the point of discontinuous derivative.

    Edited by Billy T to remove three words (in the derivatives) from your first quote of me* that my "thinking ahead dyslexia" stuck in to point (1) as well as in point (2). Point (1) should have only spoke of the failure at discontinuities.
    * and in the original too.
     
    Last edited by a moderator: Dec 16, 2014
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  3. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    Yes, but I like to note that the following two functions (A) & (B) have the same identical FT:

    (A) y = 0 for x< 0. & y = 2 for x greater than OR EQUAL to 0 while x is less than OR EQUAL 1. & X = 0 for x > 1.
    Or without most of the words: y=0 for x < 0. y= 2 for x = or > 0 but y=0 for x > 1.
    I. e. y is 2 at x = 0 & at x = 1 and all x between but 0 elsewhere.

    (B) y = 0 for x< 0 OR EQUAL to 0. & y = 2 for 1 > x > 0. & y = 0 for x equal to 1 or greater.
    I. e. y is 0 at x = 0 & at x = 1 and y =2 at points all x between, but 0 elsewhere.

    I think the infinite FT takes the value 1 for y, at x=0, and x =1. Not either case (A)'s 2 nor case (B)'s 0; but be that as in may the FT does NOT perfectly reproduce either function (A) or (B) - that was my point (1).

    My point (2) that the FT also fails if there is only a difference between the right and left limiting slopes* of a curve at some point I now think is wrong. Thanks for correcting me with example at end of your post.

    * The square wave being the extreme case. At the corners the slope changes between zero and infinity.
     
    Last edited by a moderator: Dec 16, 2014
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  5. arfa brane call me arf Valued Senior Member

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    The two squares are a diagram of two separate parts of a knot (or its shadow). You connect these figures (which are symbolic representations) with a pair of lines to show that the two parts are connected.

    This seems like something really obvious, except for why there isn't just one connecting line between the squares. The answer to that question is the Jordan Curve Theorem.
    Except that your method isn't very general; I gave a method that works with convex figures, and said that it isn't general. I thought, perhaps understanding why these methods aren't general and what to do about it might give some insight into why the Jordan Curve Theorem is difficult to prove (do you know?).
    I think you need to support this claim with a lot more. Consider how easy it is to "well-define" a set of points which is the boundary of a closed convex figure such as a circle or a square. It gets more difficult if the figure isn't convex and has a lot of bends in it (here, I'm not bothering to define "bend" but leaving it up to you).
     
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  7. rpenner Fully Wired Valued Senior Member

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    \(A(x) = \left\{ \begin{array}{lcl} 0 & \quad \quad \quad & \textrm{if} \; x \lt 0 \\ 2 & & \textrm{if} \; 0 \leq x \leq 1 \\ 0 & & \textrm{if} \; 1 \lt x \end{array} \right.\)
    \(B(x) = \left\{ \begin{array}{lcl} 0 & \quad \quad \quad & \textrm{if} \; x \leq 0 \\ 2 & & \textrm{if} \; 0 \lt x \lt 1 \\ 0 & & \textrm{if} \; 1 \leq x \end{array} \right.\)
    These are discontinuous functions. As per my post #131 they both have the reconstruction:
    \(C(x) = \left\{ \begin{array}{lcl} 0 & \quad \quad \quad & \textrm{if} \; x \lt 0 \\ 1 & & \textrm{if} x = 0 \\ 2 & & \textrm{if} \; 0 \lt x \lt 1 \\ 1 & & \textrm{if} x = 1 \\ 0 & & \textrm{if} \; 1 \lt x \end{array} \right.\)

    This is not the Fourier transform, it is the inverse Fourier transform of the Fourier transform of either A or B.

    But
    \(D(x) = A(x) - B(x) = \left\{ \begin{array}{lcl} 2 & \quad \quad \quad & \textrm{if} \; x \in \{ 0, 1 \} \\ 0 & & \textrm{if} \; x \not\in \{ 0, 1 \} \end{array} \right.\)
    is one of those functions similar to what I described above -- only a finite number of points have non-zero, finite values.
    Thus \(\forall u,v \in \mathbb{R} \quad \int_u^v D(x) \, dx = 0\) so it's reconstruction is \(E(x) = 0\).
    The Fourier transform is not magic, it is blind to the same type of features that the operation of integration is invisible to.

    \(\int_{-\infty}^{\infty} A(x) \, dx \; = \; \int_{-\infty}^{\infty} B(x) \, dx \; = \; \int_{-\infty}^{\infty} C(x) \, dx \; = \; 2\)
    Expecting more is unreasonable. (Because of the fundamental theorem of calculus.)

    I think you will find that the Gibbs phenomenon is closer to how Guest256 describes it in that the Gibbs phenomenon goes away with adding the infinite number of terms. The phenomena of not matching arbitrary discontinuities is different as the Gibbs phenomenon was a slowness of convergence in the neighborhood of non-removable discontinuities.
    The language I was taught was it has "no slope" . But since a function can only have one value at a point, calling it discontinuous is a bit better than saying "infinite slope."
     
    Last edited: Dec 16, 2014
  8. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    You seem to be confirming my idea that the FT of the FT would take the value 1 not 2 or 0 at the discontinuity. I. e. pick the mid value of the step (why I made the non-zero values of my two functions be 2 (and not 1)

    Sorry it is late in Brazil (my bed time) and I am too rusty on proper math notations to follow rest of your post.
    Also sorry I can't even copy your nice tex display properly - I never use tex.
    PS my 2nd try at least got the first part better (briefly, but now it is messed up to).
     
    Last edited by a moderator: Dec 17, 2014
  9. arfa brane call me arf Valued Senior Member

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    Hmm. This idea of projecting rays from a point has legs, in that it makes the centre of a circle just another projection point.

    What we can see is that every point in the plane has a complete circle of directions (to project rays outward) over it. This is what's called the fiber over the space; a fiber is a set of directions over each point in the space, which is the Euclidean plane in this case (where we draw curves and symbols).

    So obviously, since you can map (via projections) every point in the interior of a circle to its boundary (or a section of the boundary), you can show that the disc (a surface of 2 dimensions) has the same cardinality as the boundary (a surface of 1 dimension). Scary, huh?

    An obvious thing about drawing curves or polygons: you can draw a continuous "smooth" curve without stopping, when you draw a polygon or a polygonal chain you stop and restart at each vertex. Drawing figures on a plane surface says something about the mathematics. What do you suppose it is?
     
  10. arfa brane call me arf Valued Senior Member

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    Well, I've been inspired by the phrase "network topology" and come up with the following:

    When you cross a loop of some material like wire or string over itself the important feature is the crossing, and that there are two ways to arrange this in three dimensions. You focus on the "X" and can "redraw" the rest of the loop as a pair of boxes connected to this X shape; untwisting the X leaves whatever is in the boxes unchanged (it will rotate part of the loop and any local crossings in that part).

    The crossing is what you have when you spatially transpose a pair of uncrossed lines connecting the boxes, hence is analogous to a 2-input, 2-output switch; precisely a crossbar switch.

    One important detail is that crossbar switches are most likely manufactured to switch in one of the two possible ways (that two tracks can cross), but you can imagine that two kinds of switch are available and identified as such (like type A or B, say). Or you could have a switch that mirrors itself, a switch with parity so the input (or switch state) can be +1 or -1 when "on" , and 0 means "off". The information about which track crosses which isn't needed by the switching function or a network of switches (since switching is the important thing, not the mechanism).

    This information is however, important if the problem is determining if a loop or loops is knotted in 3-space.

    To make a closed loop with a 2x2 crossbar switch or a series of them, you connect inputs and outputs together (this is equivalent to closing a braid).
    Exercise: construct a closed loop "figure eight" with a single 2x2 crossbar switch that has only one "on" mode. How many switches in a row are needed to make a linked pair of loops?

    p.s. the above exercise can be carried out with a pen and paper.
     
    Last edited: Dec 20, 2014
  11. river

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    The crossing is interesting , since it results in three dimensions

    Only if though, the wire or string , when crossed , can be expanded within our space

    They both can be

    It seems obvious that they should and do

    Whats interesting is that then two dimensional thinking about the Universe is irrelevant
     
  12. arfa brane call me arf Valued Senior Member

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    Kauffman on the above: "Reidemeister showed that two knots or links in three-dimensional space can be deformed continuously one into the other (the usual notion of ambient isotopy) if and only if any diagram (obtained by projection to a plane) of one link can be transformed into a diagram for the other link via a sequence of Reidemeister moves. Thus these moves capture the full topological scenario for links in three-space."

    Hence, given there is a sequence of moves taking K to K', you have something strongly (stringly) analogous to an algorithmic structure in this space, as well as the peculiar winding (embedding of a circle). Double hence, the use of a switching network isn't that far away from it all, besides, the 2x2 crossbar switch, with slight modifications, is a useful model. At least I think so.

    You do have to remember that a 2x2 crossbar switch "breaks" a pair of tracks, and "reconnects" them, analogous to a cutting and splicing operation.
     
    Last edited: Dec 20, 2014
  13. arfa brane call me arf Valued Senior Member

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    Someone made some claim about the topology of a point, which was shut down by QuarkHead.

    A point in \( \mathbb R \) can have only two directions "over" it; a point in \( \mathbb R^n, n \ge 2 \) has an infinite set of directions over it.

    So in \( \mathbb R^2 \), take a subset of the total set of directions over some point, p.
    Assign rays to this subset, if the subset is continuous the rays must pass through a set of points in the neighbourhood of p which is a continuous 1-dimensional curve.

    Therefore if the total set of directions over p is continuous, the set of rays from p associated to that set of directions must pass through a continuous closed curve such that p is in the interior of this curve. The closed curve has a completely arbitrary "shape" (uh oh!).

    Every point in the interior has the same 'status' as p, or mathematically, occupies the same place.
     
    Last edited: Dec 25, 2014
  14. arfa brane call me arf Valued Senior Member

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    Actually from a point p, the set of points in some continuous curve that rays from p intersect can't be that arbitrary, at least not if you want the rays to intersect only one point each, and not be tangent to the curve anywhere. What if those conditions aren't considered?

    Anyways, back to this idea of switching a pair of tracks, and the connection to knot theory.
    Obviously there are two types of switch. Kauffman distinguishes them by defining an abstract region on the left of an undercrossing line as an observer moves along it towards the crossing, calling this region "A".

    You could also call this an "A-type" switch which has the (corresponding part of the) overcrossing track on the left of the undercrossing track for an observer moving along the latter towards the crossing.

    You can see that no matter how a crossing is rotated (in three dimensions), this relation doesn't change (as long as the abstract "region" or surface stays connected to both lines); it's an invariant.

    More correctly, if the rest of the knot as two abstract boxes connected to the two lines is left unchanged, the "movement towards a crossing" relation is a kind of knot symmetry.

    And you can see that a B-type switch is just an A-type with everything rotated 90 degrees. Now an observer moving along the undercrossing line (track) will see the overcrossing line on their right, and the region is therefore a B region. What is harder to see is that \( A \) and \( B \) are like functions, and that \( B = A^{-1} \).
     
    Last edited: Dec 27, 2014
  15. arfa brane call me arf Valued Senior Member

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    So whether or not the original question has had anything approaching "the" answer offered in this thread, knots are a kind of poster child of the subject, and knots are interesting by themselves.

    Questions like, how are knots, or the mathematics involved, related to such subjects as special relativity, or quantum fields? What kinds of relations are there anyway, between knotted or linked circles and physics?

    What do minimal or Seifert surfaces have to do with a switching function, how are we supposed to reconcile a static loop, something we can model with wire, with dynamical systems like switching networks?

    I don't really know the answers to any of that. I'm still trying to understand what Kauffman's bracket polynomials say about the crossings you can make in simple loops of wire. You do have to start with a loop that has no crossings, that is, no crossings in any projection on to a plane.
    Then it's fairly obvious that there are a possibly infinite number of ways to deform a flat, circular loop of wire so a projection of it has one crossing. What about the minimal surface? It deforms as well (if there is one).

    The crossing then is defined by this surface and a direction towards the crossing, along the undercrossing segment.
    Ok, but if you bend the wire loop again so the projection looks like a loop inside a loop you invert the crossing and the minimal surface changes to a thinner band.

    Kauffman's bracket doesn't distinguish between these shapes! the same polynomial describes two different ways to connect a pair of loops with a single crossing, this is because adding a loop to a loop can be done with one inside (resp. outside) the other, and no intersections.

    So . . .
     
  16. Farsight

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    I don't think there's much of a tie in between knots and special relativity. But in QFT we start with the photon field, and the electron field. And IMHO what TQFT is saying, is that the electron field is the photon field when it's tied in a knot. Other people might beg to differ, but see post #5 in this thread where rpenner said this:

    "With the proton, we start with the up and down quarks, each with 4 quantum fields, and we add the 16 gluon fields for a total of 24 quantum fields or 47 degrees of freedom and we are still a long way from describing the electromagnetic and mass properties of the proton. So in summary, a quantum bound state is a little like a high-dimensional knot of quantum field configurations that happens to be an energy eigenstate which says the state doesn't change over time in any fundamental way. "

    IMHO a massive particle is a knot of field. Look at the blue trefoils here. Trace round clockwise from the bottom left, and the crossing over directions are up down up. Now where have we heard that before?

    I don't know. But you start with the trivial knot and torus animations:

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    I don't think wire sounds like the right place to start. We make electrons and positrons out of photons in pair production. A photon isn't a piece of wire. It's more like a wave in a wire, like a guitar string. But this wire is but one wire in a lattice of wires representing space. Only space isn't some lattice of wires.
     
    Last edited: Jan 5, 2015
  17. PhysBang Valued Senior Member

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    Why do you poison every thread with this nonsense that you can't support, other than by finger painting?
     
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  18. arfa brane call me arf Valued Senior Member

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    .
    I think wire is a perfectly good place to start. Drawing figures in the plane isn't a bad place to start either, here, you have to think about how to 'encode' the kinds of things you can do in three dimensions (to a piece or a loop of wire) in two dimensions; that is, how to "recover" the three dimensional information lost in a projection.

    Kauffman says this about spacetime:
    "What does it mean to produce a combinatorial model of spacetime? One thinks of networks, graphs, formal relationships, a fugue of interlinked constructions. Yet what will be fundamental to such an enterprise is a point of view, or a place to stand, from which the time space unfolds.

    Before constructing, before speaking and even before logic stands a simple concept, and that is the concept of distinction. I take this concept first in ordinary language where it has a multitude of uses and a curious circularity. For there can be no seeing without a seer, and no speaking without a speaker.

    We, who would discuss this concept of distinction, become distinguished . . .
    Distinction is a concept that requires its own understanding. Fortunately, there are examples: Draw a distinction; draw a circle in the plane; delineate the boundary of this room; indicate the elements in the set of prime integers.
    . . .
    At the risk of creating a very long digression, I want to indicate how the Lorentz group arises naturally in relation to any distinction. "

    ( . . . goes on to show how lightcone coordinates and the principle of relativity lead to a minimal representation of any spacetime point as an Hermitian 2x2 matrix, then to analysing the "distinctions" that "build" this structure).
     
  19. arfa brane call me arf Valued Senior Member

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    The type I Reidemeister move changes the writhe of a knot or link. What that means is, making a figure 8 out of a loop is just a type I move, viz:

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    Here the diagram is meant to be part of a larger one.

    So you can repeat this move to make as many crossings as you like (all "undo-able" hence not adding any "knottedness" to the link). The trefoil has a writhe of +3 (the writhe is just the sum over crossings of the sign of each crossing, + or -). So you can give the trefoil a writhe of 0 by adding three negative type I Reidemeister moves.

    Although the writhe is an invariant of regular (planar) isotopy, obviously the type I move is a problem. The right hand diagram above (the "result" of twisting a loop over itself) has the same bracket polynomial as a figure 8, which is because the free ends can be connected together in two distinct ways, one way you get a figure 8, the other way it's like (the boundary of) a twisted annulus. Suppose there is a minimal surface inside the boundary of the right hand diagram (figure!), what happens to it as you connect the free ends in either of the two possible ways? Wouldn't it 'need' to flow out and connect to the rest of the boundary?

    Note that the type I move is very similar to what an elastic band does when you twist it sufficiently. Then, consider that if the above diagram(s) were of embedded bands, the right hand one would be twisted around itself if it was stretched apart.

    This probably sounds a bit prosaic; rubber bands, soap films as minimal surfaces. If you're interested in what kind of physics you can talk about with either I'm sure there's any amount of videos on youtube.

    Maybe I should expand on things a bit: the type II and III Reidemeister moves (I'll call these moves instead), do not change the writhe of a knot

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    The type II move adds one + and one - crossing;

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    likewise for the type II move. So the type I move adds (or removes) either a + or a - to the writhe of a knot or link . . .
     
    Last edited: Jan 7, 2015
  20. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    I have not been following, but before I try, a simple question about the right side with loop:

    Does one part of the loop area touch the other making a doubly connected domain? Or is it still a singly connected domain - I. e. topographically a doughnut with two long cylindrical tubes attached?
     
  21. Farsight

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    • This post appears to be off-topic for this thread.
    There's a bit of a problem with that. I'll demonstrate in a minute.

    There's a problem there too. Spacetime is static. Not dynamical. It's an abstract thing which models all times at once. To depict it we drop one of the space dimensions to leave a plane. Then the time dimension is depicted vertically, and we draw worldlines in the block. But now it's a block universe. The wordline represents the position of a body at all times. The body doesn't move up the worldline. There is no motion in spacetime. The motion is through space.

    Another problem is that the electron is "just field". It doesn't have a surface, just as a whirlpool doesn't have an outside edge.

    A light cone is just an abstract thing in a static mathematic "space" called spacetime. Sorry to pour cold water on everything you've raised, but IMHO there are a lot of distractions, and it's important to avoid them.
     
  22. Farsight

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    I said I'll demonstrate a bit of a problem. Here it is:

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    That's got the writhe. It's a figure 8. Yes. Look at this picture. Look at the black line. This image is from the same paper. So we've got a loop that's also a double loop. Rather like a Moebius strip. See Dirac's belt on Mathspages:

    "The un-twisted belt is called an orientable surface, because a right-handed figure (for example) remains right-handed, regardless of how it is translated around the manifold. In contrast, the Mobius strip is a non-orientable surface, because a right-handed figure, moved continuously around the loop until arrive back at its starting point, becomes left-handed. An object must be translated around the loop twice in order to be restored to its original position and chirality. In this sense a Mobius strip is reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state."
     
  23. PhysBang Valued Senior Member

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    That is not a problem. That is a series of unconnected facts. Please produce an argument.
     

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