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What Is Knot Theory? Why Is It in Mathematics? [pdf] (osaka-cu.ac.jp)
94 points by _of on Aug 9, 2016 | hide | past | web | favorite | 35 comments



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


That is awesome!


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.


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.


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.


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.


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 [2] https://www.youtube.com/watch?v=zd_HGjH7QZo [3] http://facstaff.susqu.edu/brakke/polycut/polycut.htm


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

[2] in memory of Zeeman - http://www2.warwick.ac.uk/knowledge/science/zeeman


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.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.


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


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/PWsept...


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.


Loops tangle creating more loops easierly and do not undo themselves typically but rather get tighter upon pulling


that's knot funny


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)?


I think it comes down to the difficulty of finding such things. The goal is usually to find "enough" invariants that all objects can be distinguished, but we settle for just whatever invariants we can happen to find. Finding an invariant is pretty easy: it's straight-forward to show that something is invariant under a certain transformation since you just apply a transformation (in this case, usually a homotopy or similar) and see what it can do to the invariant. For knot theory, you can often check just what it does under the Reidemeister moves, giving you just a couple things to check.

How would you go about showing that something is a dinvariant? You would need to take two objects that are distinct and show that their dinvariants are distinct. I.e. use the fact that a homotopy between the two knots does not exist. This is much more difficult, though it does get done. In particular, it's done whenever we have a complete classification of all objects of a certain type (say orientable surfaces, classified by the single invariant (and also dinvariant) their genus). Though generally here we would be actually approaching this from an invariant perspective but just showing that once you have enough invariants, they collectively become a dinvariant.


Invariants are useful because they allow you to distinguish equivalent knots, which is otherwise really hard to do. If two knots have different invariants, then they are surely different.

Dually, you would want the following: if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function, and that makes them mostly useless.

(full disclosure: lawyer who did his undergraduate degree in math, not a topologist)


> Invariants are useful because they allow you to distinguish equivalent knots, which is otherwise really hard to do. If two knots have different invariants, then they are surely different.

> if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function, and that makes them mostly useless.

That doesn't make them useless. Dinvariants just serve a different purpose:

- invariants serve the purpose of distinguishing knots that are different

- dinvariants serve the purpose of detecting that knots that look very different are actually the same


That is extremely difficult but that is, indeed, the goal. The main business of algebraic topology is doing exactly that.


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?


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)


> 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. :-)


Oh! shit, I misunderstood your proposal completely. That is definitely not the goal of Algebraic Topology... Sorry about that.

If you say two knots are equivalent, in mathematics, that means they are equal, one and the same. You can't have a tool that assigns different structures to the same thing. But I see where you are coming from. In mathematics we have the concept of a presentation (of a group or a vector space, for example). You can imagine a presentation of a knot like in a two dimensional representation (typical one) and assign an invariant to that. That wouldn't be an invariant of a knot, that would be an invariant of a knot, as you mention, but it would be an invariant of a knot plus some extra structure. Those objects are also studied in many instances in which the original is two difficult. As simple example imagine a knot plus an orientation. Invariants of that structure might be dinvariants of the knot.


> If you say two knots are equivalent, in mathematics, that means they are equal, one and the same.

As far as I know a knot is a continuous embedding of S^1 into R^3. Equal means exactly same subset of R^3. Equivalence of knots means that there is a suitable ambient isotopy between the nots (I just looked this up in wikipedia: https://en.wikipedia.org/wiki/Knot_theory#Knot_equivalence). So equality is a much stronger criterion than equivalence.

So 'equal => equivalence', but the other direction 'equivalence => equal' trivially does not hold (just apply some transition in R^3; these knots are equivalent but not equal).


No, there is only one unknot and only one 3_1 knot, for example. Am embedding is not a knot, the knot is the embedding + the equivalence relation. So, just the embedding would be a presentation of the knot.


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?


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.


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 :]


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.


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


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


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



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


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




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