
What Is Knot Theory? Why Is It in Mathematics? [pdf] - _of
http://www.sci.osaka-cu.ac.jp/~kawauchi/WhatIsKnotTheory.pdf
======
cottonseed
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
That is awesome!

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dkarapetyan
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
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
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
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.

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roywiggins
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)

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tupilaq
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)

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RachelF
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...](https://www.sciencedaily.com/releases/2016/02/160210170411.htm)

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
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)

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wolfgke
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)?

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Varinius
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)

~~~
wolfgke
> 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

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imh
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?

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wmsiler
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.

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

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xaox
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)

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

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pizza
also
[https://en.wikipedia.org/wiki/Racks_and_quandles](https://en.wikipedia.org/wiki/Racks_and_quandles)

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

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

