8 ms·

What Is Knot Theory? Why Is It in Mathematics? [pdf]

94 points by _of 10y ago

RachelF 10y ago
In the late 1800's knot theory was quite popular with physicists. Now there is the much bigger string theory: https://www.sciencedaily.com/releases/2016/02/160210170411.htm https://www.sciencedaily.com/releases/2016/02/160210170411.h... On a lighter note I could use some knot theory to explain why earphone or computer cables always seem to tie themselves up, despite my best efforts to keep them apart.
- emptybits 10y ago
  Here's a paper on the topic: "Spontaneous knotting of an agitated string" [1] From the abstract: "We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots." [1] http://www.ncbi.nlm.nih.gov/pubmed/17911269 http://www.ncbi.nlm.nih.gov/pubmed/17911269
- cohomologo 10y ago
  Knot theory is still quite popular with physicists of certain sorts, such as condensed matter physicists that study "topological phases" - a popular account that I like can be found here: https://www-thphys.physics.ox.ac.uk/people/SteveSimon/PWsept10.pdf https://www-thphys.physics.ox.ac.uk/people/SteveSimon/PWsept...
- NolF 10y ago
  There are fewer desired unknot states to your cable than there are tangled and knotted states to them. There also is some confirmation bias as you are less likely to noticed the desired state versus having to battle through the knots for 5 minutes.
- pizza 10y ago
  Loops tangle creating more loops easierly and do not undo themselves typically but rather get tighter upon pulling
- triplesec 10y ago
  that's knot funny
tupilaq 10y ago
hey, I was just reading about Christopher Zeeman [1] who was quite the topologist. He founded the Maths Faculty at the University of Warwick [2]. Maths is a truly astonishing discipline begging the question, are we all just made of maths? Zeeman was very quotable: "Technical skill is mastery of complexity while creativity is mastery of simplicity." [1] Wikipedia entry - https://en.wikipedia.org/wiki/Christopher_Zeeman https://en.wikipedia.org/wiki/Christopher_Zeeman [2] in memory of Zeeman - http://www2.warwick.ac.uk/knowledge/science/zeeman http://www2.warwick.ac.uk/knowledge/science/zeeman
alexilliamson 10y ago
Topology has always seemed like the exotic yet substantive branch of mathematics... would love to see more reading like this posted.
dkarapetyan 10y ago
I think a lot of human understanding is basically intuition about topological invariants in various "spaces". If you go around asking famous thinkers what they see when they think they all describe similar kinds of imagery, fuzzy shapes that merge and unmerge in various ways as they probe the subject. One good book I've found on the subject is https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From.
- catnaroek 10y ago
  Intuition quickly becomes unreliable when you move to spaces with weird topologies, like non-Hausdorff and non-(pseudo)metrizable spaces. When your intuition stops being useful, you actually need to calculate.
  - dkarapetyan 10y ago
    I suspect a few folks that studied p-adic numbers extensively would disagree. And in general topologists and algebraists that study non-euclidean things in general.
    - catnaroek 10y ago
      The p-adic numbers can be equipped with a metric. The induced topology isn't Euclidean, but it's pretty tame compared to what you can see in a general topological space.
sajid 10y ago
I find it remarkable that it took humans so long (1849) to start mathematically investigating something as fundamental as knots.
- sushid 10y ago
  Paper folding is something just as fundamental yet it wasn't until 1893 that humans started mathematically analyzing it.
xaox 10y ago
Reference: Kawauchi, Akio, and Tomoko Yanagimoto. "What Is Knot Theory? Why Is It In Mathematics?." In Teaching and Learning of Knot Theory in School Mathematics, pp. 1-15. Springer Japan, 2013. http://link.springer.com/chapter/10.1007/978-4-431-54138-7_1 http://link.springer.com/chapter/10.1007/978-4-431-54138-7_1
roywiggins 10y ago
Knots can also be identified with spaces that don't seem "knotty" at first glance. See Thurston's Knots to Narnia[1] and Not Knot[2]. PolyCut[3] is a little applet that can visualize these knotty portals. [1] https://www.youtube.com/watch?v=IKSrBt2kFD4 https://www.youtube.com/watch?v=IKSrBt2kFD4 [2] https://www.youtube.com/watch?v=zd_HGjH7QZo https://www.youtube.com/watch?v=zd_HGjH7QZo [3] http://facstaff.susqu.edu/brakke/polycut/polycut.htm http://facstaff.susqu.edu/brakke/polycut/polycut.htm
imh 10y ago
Why is it just in 3D? 1D things making knots in 3D seems like it would have immediate analogs for m-dimensional things in n dimensions. Is that not the case? Is it not as rich an area of study or something?
- detaro 10y ago
  AFAIK, the term "knot" is only used for 1D things, and those are not very interesting in higher dimensions because you can always unravel them. The more general field is called (Geometric) Topology, and includes all your m-dimensional objects in n-dimensional spaces.
- aznstriker24 10y ago
  The following is hand-wavey, but only because I am not equipped to give an actual explanation :] Essentially, knots (i.e. 1D things embedded into some other space without crossing itself) are trivial in dimensions lower than three because we don't have enough space to make it interesting. A 1D thing that doesn't cross itself in the plane must be a circle, warped in some way, but it can be unwarped without ripping the plane. On the other hand, knots in dimension higher than three are trivial because we have too much space to work with. I don't know the details here, but this is what I've been told. It's kinda miraculous that knots in three dimensions have such interesting and rich structure. Hopefully an expert will come by and give a more detailed explanation, but in the meantime, hope this helps :]
- wmsiler 10y ago
  You are right that interesting knot theory does exist in higher dimensions. It is appropriately called higher dimensional knot theory. It considers spheres of dimension m embedded in n-dimensional space. When m = 1, you get a 1-dimensional sphere, which is a circle. There are restrictions on which m and n yield interesting math. Intuitively, if the dimension of the sphere is too small compared to the ambient space (i.e. m is much smaller than n), then there will be so much wiggle room, that any knot can be "untied" without crossing itself (i.e. every knot is the trivial unknot). If the dimension of m is too big compared to n, then there is not enough room to twist things around, and so again nothing can get knotted. It's been a while since I've studied this, but I believe it's the case that the only time you get nontrivial knots is when n = m + 2. The most well known case, of course, is when m = 1 and n = 3. But for every value of m >= 1, there are nontrivial knots in dimension m + 2. I believe it is indeed a rich area of research.
cottonseed 10y ago
One of my favorite mathematical diagrams: http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf, from Matveev, Fomenko, Algorithmic and computer methods in three-dimensional topology.
- jeffwass 10y ago
  That is awesome!
kixpanganiban 10y ago
Here's a good video from Numberphile: https://www.youtube.com/watch?v=aqyyhhnGraw https://www.youtube.com/watch?v=aqyyhhnGraw
wolfgke 10y ago
One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants. But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different simplical complexes or different CW complexes), the dinvariant will assign different values. On the other hand, if the knots are equivalent, the assigned values might not be equal. What I want to know is: Why doesn't there seem to exist a theory of dinvariants for knots (or topological spaces)?
- nhatcher 10y ago
  That is extremely difficult but that is, indeed, the goal. The main business of algebraic topology is doing exactly that.
  - wolfgke 10y ago
    As far as I understand it the far goal is to find invariants that are also dinvariants. But why don't we build a systematic theory of dinvariants (similar to the theory of invariants) to get a much better understanding of them?
    - nhatcher 10y ago
      Because we can't find them :S. In the case of knots, we would rather have polynomials associated with knots that satisfy your requirement. But we just don't know how to do that. It is the same with topological spaces and homotopy theory or cohomology. Whenever somebody finds a new algebraic invariant that is capable of differentiating between simmilar structures that's a big deal. Like the Jones polynomials did (Or the Donaldson polynomials in the 80's for four manifolds)
      - wolfgke 10y ago
        > Whenever somebody finds a new algebraic invariant that is capable of differentiating between simmilar structures that's a big deal. Not every dinvariant needs to be an invariant. :-)
pizza 10y ago
also https://en.wikipedia.org/wiki/Racks_and_quandles https://en.wikipedia.org/wiki/Racks_and_quandles

94 points by _of 10y ago

In the late 1800's knot theory was quite popular with physicists. Now there is the much bigger string theory: https://www.sciencedaily.com/releases/2016/02/160210170411.htm https://www.sciencedaily.com/releases/2016/02/160210170411.h... On a lighter note I could use some knot theory to explain why earphone or computer cables always seem to tie themselves up, despite my best efforts to keep them apart.

emptybits 10y ago
Here's a paper on the topic: "Spontaneous knotting of an agitated string" [1] From the abstract: "We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots." [1] http://www.ncbi.nlm.nih.gov/pubmed/17911269 http://www.ncbi.nlm.nih.gov/pubmed/17911269
cohomologo 10y ago
Knot theory is still quite popular with physicists of certain sorts, such as condensed matter physicists that study "topological phases" - a popular account that I like can be found here: https://www-thphys.physics.ox.ac.uk/people/SteveSimon/PWsept10.pdf https://www-thphys.physics.ox.ac.uk/people/SteveSimon/PWsept...
NolF 10y ago
There are fewer desired unknot states to your cable than there are tangled and knotted states to them. There also is some confirmation bias as you are less likely to noticed the desired state versus having to battle through the knots for 5 minutes.
pizza 10y ago
Loops tangle creating more loops easierly and do not undo themselves typically but rather get tighter upon pulling
triplesec 10y ago
that's knot funny

tupilaq 10y ago

hey, I was just reading about Christopher Zeeman [1] who was quite the topologist. He founded the Maths Faculty at the University of Warwick [2]. Maths is a truly astonishing discipline begging the question, are we all just made of maths? Zeeman was very quotable: "Technical skill is mastery of complexity while creativity is mastery of simplicity." [1] Wikipedia entry - https://en.wikipedia.org/wiki/Christopher_Zeeman https://en.wikipedia.org/wiki/Christopher_Zeeman [2] in memory of Zeeman - http://www2.warwick.ac.uk/knowledge/science/zeeman http://www2.warwick.ac.uk/knowledge/science/zeeman

alexilliamson 10y ago

Topology has always seemed like the exotic yet substantive branch of mathematics... would love to see more reading like this posted.

dkarapetyan 10y ago

I think a lot of human understanding is basically intuition about topological invariants in various "spaces". If you go around asking famous thinkers what they see when they think they all describe similar kinds of imagery, fuzzy shapes that merge and unmerge in various ways as they probe the subject. One good book I've found on the subject is https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From.

catnaroek 10y ago
Intuition quickly becomes unreliable when you move to spaces with weird topologies, like non-Hausdorff and non-(pseudo)metrizable spaces. When your intuition stops being useful, you actually need to calculate.
- dkarapetyan 10y ago
  I suspect a few folks that studied p-adic numbers extensively would disagree. And in general topologists and algebraists that study non-euclidean things in general.
  - catnaroek 10y ago
    The p-adic numbers can be equipped with a metric. The induced topology isn't Euclidean, but it's pretty tame compared to what you can see in a general topological space.

sajid 10y ago

I find it remarkable that it took humans so long (1849) to start mathematically investigating something as fundamental as knots.

sushid 10y ago
Paper folding is something just as fundamental yet it wasn't until 1893 that humans started mathematically analyzing it.

xaox 10y ago

Reference: Kawauchi, Akio, and Tomoko Yanagimoto. "What Is Knot Theory? Why Is It In Mathematics?." In Teaching and Learning of Knot Theory in School Mathematics, pp. 1-15. Springer Japan, 2013. http://link.springer.com/chapter/10.1007/978-4-431-54138-7_1 http://link.springer.com/chapter/10.1007/978-4-431-54138-7_1

roywiggins 10y ago

Knots can also be identified with spaces that don't seem "knotty" at first glance. See Thurston's Knots to Narnia[1] and Not Knot[2]. PolyCut[3] is a little applet that can visualize these knotty portals. [1] https://www.youtube.com/watch?v=IKSrBt2kFD4 https://www.youtube.com/watch?v=IKSrBt2kFD4 [2] https://www.youtube.com/watch?v=zd_HGjH7QZo https://www.youtube.com/watch?v=zd_HGjH7QZo [3] http://facstaff.susqu.edu/brakke/polycut/polycut.htm http://facstaff.susqu.edu/brakke/polycut/polycut.htm

imh 10y ago

Why is it just in 3D? 1D things making knots in 3D seems like it would have immediate analogs for m-dimensional things in n dimensions. Is that not the case? Is it not as rich an area of study or something?

detaro 10y ago
AFAIK, the term "knot" is only used for 1D things, and those are not very interesting in higher dimensions because you can always unravel them. The more general field is called (Geometric) Topology, and includes all your m-dimensional objects in n-dimensional spaces.
aznstriker24 10y ago
The following is hand-wavey, but only because I am not equipped to give an actual explanation :] Essentially, knots (i.e. 1D things embedded into some other space without crossing itself) are trivial in dimensions lower than three because we don't have enough space to make it interesting. A 1D thing that doesn't cross itself in the plane must be a circle, warped in some way, but it can be unwarped without ripping the plane. On the other hand, knots in dimension higher than three are trivial because we have too much space to work with. I don't know the details here, but this is what I've been told. It's kinda miraculous that knots in three dimensions have such interesting and rich structure. Hopefully an expert will come by and give a more detailed explanation, but in the meantime, hope this helps :]
wmsiler 10y ago
You are right that interesting knot theory does exist in higher dimensions. It is appropriately called higher dimensional knot theory. It considers spheres of dimension m embedded in n-dimensional space. When m = 1, you get a 1-dimensional sphere, which is a circle. There are restrictions on which m and n yield interesting math. Intuitively, if the dimension of the sphere is too small compared to the ambient space (i.e. m is much smaller than n), then there will be so much wiggle room, that any knot can be "untied" without crossing itself (i.e. every knot is the trivial unknot). If the dimension of m is too big compared to n, then there is not enough room to twist things around, and so again nothing can get knotted. It's been a while since I've studied this, but I believe it's the case that the only time you get nontrivial knots is when n = m + 2. The most well known case, of course, is when m = 1 and n = 3. But for every value of m >= 1, there are nontrivial knots in dimension m + 2. I believe it is indeed a rich area of research.

cottonseed 10y ago

One of my favorite mathematical diagrams: http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf, from Matveev, Fomenko, Algorithmic and computer methods in three-dimensional topology.

jeffwass 10y ago
That is awesome!

kixpanganiban 10y ago

Here's a good video from Numberphile: https://www.youtube.com/watch?v=aqyyhhnGraw https://www.youtube.com/watch?v=aqyyhhnGraw

wolfgke 10y ago

One question if some knot theorists (or at least topologists) are reading along: I can understand why knot theorists are so interested in finding invariants. But now let's define a "dinvariant" ("dual invariant" or "different invariant"): A dinvariant assigns to each knot also some object such that if the knots are different (or topological space are different in their class where they come from (say: are different simplical complexes or different CW complexes), the dinvariant will assign different values. On the other hand, if the knots are equivalent, the assigned values might not be equal. What I want to know is: Why doesn't there seem to exist a theory of dinvariants for knots (or topological spaces)?

nhatcher 10y ago
That is extremely difficult but that is, indeed, the goal. The main business of algebraic topology is doing exactly that.
- wolfgke 10y ago
  As far as I understand it the far goal is to find invariants that are also dinvariants. But why don't we build a systematic theory of dinvariants (similar to the theory of invariants) to get a much better understanding of them?
  - nhatcher 10y ago
    Because we can't find them :S. In the case of knots, we would rather have polynomials associated with knots that satisfy your requirement. But we just don't know how to do that. It is the same with topological spaces and homotopy theory or cohomology. Whenever somebody finds a new algebraic invariant that is capable of differentiating between simmilar structures that's a big deal. Like the Jones polynomials did (Or the Donaldson polynomials in the 80's for four manifolds)
    - wolfgke 10y ago
      > Whenever somebody finds a new algebraic invariant that is capable of differentiating between simmilar structures that's a big deal. Not every dinvariant needs to be an invariant. :-)

pizza 10y ago

also https://en.wikipedia.org/wiki/Racks_and_quandles https://en.wikipedia.org/wiki/Racks_and_quandles