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"In topology, knot theory is the study of mathematical knots."

This is how Wikipedia defines knot theory. I have no idea what this is supposed to mean, but it does seem interesting. The rest of the article is full of examples of knots, their notation and such, which I understand a little bit better, but I still fail to understand why and how they are studied. So, what exactly is knot theory about?

What characteristics of knots are studied, and how does it connect to the rest of mathematics? Full disclosure: I have experience with analysis, know at least the basics of abstract algebra, and I think I could understand elementary topology. So feel free to give me only a moderately technical description.

Peter Mortensen
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user3460322
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    Colin Adams wrote a book called "The Knot Book" which is a great nontechnical introduction which nevertheless covers some interesting territory. – Cheerful Parsnip Nov 16 '16 at 02:19
  • I can't speak to knot theory in particular, but I would assume the big ideas are something like group theory. Can we classify all groups? Can we find ways in which a group is, in some sense, "made up of" smaller, easier to understand groups? What does it mean to understand a group, any way? Given two groups that seem like they might be "the same," how can we tell if they really are? Not all mathematicians like to play taxonomist, but for any given collection of mathematical objects, there will be those who want to categorize. – pjs36 Nov 16 '16 at 02:26
  • community wiki maybe? –  Nov 16 '16 at 02:34
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    There is an interesting use of von Neumann algebras in knot theory by Jones for which he obtained a fields medal. However I guess vN algebras are out of your scope. – Sebastian Bechtel Nov 16 '16 at 06:49
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    I can tell you what it's *knot* about... :P – mbomb007 Nov 16 '16 at 17:20

5 Answers5

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A knot, for our purposes, is a (well-behaved) "loop" in 3-dimensional space. Mathematically speaking, we could think of a knots as (injective, differentiable) functions from the unit circle to $\Bbb R^3$ (or equivalently, the image of this function in $\Bbb R^3$). Without losing any real structure, we'll suppose that these loops fit inside the sphere of radius $1$.

That's the easy part. Now, the tricky bit: what does it mean for two knots to really be the "same knot"? Intuitively, we'd like two knots to be the "same" if you can strech/squish/twist one to make the other without "tearing the rope" or passing the rope through itself. The way we encode this mathematically is to say that two knots are the same knot if they are ambient isotopic (or, for a weaker condition, ambient isomorphic). In particular, two knots $K_1,K_2 \subset B$ ($B$ is the closed unit ball) are ambient isomorphic if their complements $B \setminus K_1$ and $B \setminus K_2$ can be continuously deformed from one to the other (they are ambient isomorphic if these complements are homeomorphic).

Note: It is not enough to check whether two knots are homeomorphic, since all knots are homeomorphic to the unit circle. I believe that ambient isomorphism implies ambient isotopy in this case, but I'm not sure.

With that, the central knot theory questions are

  • How can we tell if $K_1$ is the same knot as $K_2$
  • How can we break complicated knots down into smaller knots that we understand

Another helpful way to think about knots is in terms of their knot diagrams. In particular: we take a knot, look at its projection onto a suitable plane, and keep track of all over/under crossings. It turns out that two knot diagrams correspond to the same knot if and only if one can get from one diagram to the other using Reidemeister moves.

So, how do we tell knots apart? Usually, we do so using knot invariants, properties that a knot retains no matter how exactly it's stretched, twisted, or smooshed. For example, we know that the trefoil is distinct from the "unknot" because the trefoil is tricolorable, but the unknot isn't.

Ben Grossmann
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    Some applications of knot theory are explained [here](http://mathoverflow.net/q/48222/34894), for anyone interested. – Ben Grossmann Nov 16 '16 at 02:37
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    Thanks for the answer! I now have at least a basic idea of the problems that knot theory concerns itself with, and I may take a course on it once I have some experience with topology and abstract algebra. Thank you! – user3460322 Nov 16 '16 at 14:14
  • Note: it seems that I should have said ambient *isotopic* rather than ambient isomorphic. That being said, I think the two could conditions turn out to be equivalent for knots. – Ben Grossmann Nov 16 '16 at 18:24
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    I don't know much about knot theory, but if I were to state an intuitive equivalence condition between knots, my first attempt would be something about non-self-intersecting homotopy. Is there a particular reason comparing $B\setminus K$ works better? – hmakholm left over Monica Nov 17 '16 at 09:19
  • @HenningMakholm: Yea that's my first reaction too, though I know almost nothing about knot theory. Omnomnomnom: Is there a simple intuitive reason why complement homeomorphism is equivalent to existence of non-intersecting homotopic transform? The latter clearly implies the former, but is the other direction easy to prove? – user21820 Nov 17 '16 at 10:25
  • So first of all, we should probably think in terms of ambient isotopy, which is to say that the complement of one is isotopic to the complement of the other, hence my last comment. – Ben Grossmann Nov 17 '16 at 10:39
  • To answer that question: the latter implies the former because if the loop-shaped hole self-intersects, the resulting complement will clearly have a different fundamental group, and so won't be homeomorphic to the original (I think ambient isotopy will prohibit self intersection in general, since one can make a homological argument along these lines). I don't really know why the former implies the latter, which is to say that a non self intersecting homotopy is sufficient. – Ben Grossmann Nov 17 '16 at 10:49
  • @HenningMakholm : ​ That would allow the [Bachelor's unknotting](https://books.google.com/books?id=NPxpDQAAQBAJ&pg=PA380). ​ ​ ​ ​ –  Nov 17 '16 at 14:25
  • @user21820 : ​ See my previous comment. ​ ​ ​ ​ –  Nov 17 '16 at 14:25
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    @Ricky it's not clear how that answers the question, or really what question you're trying to answer – Ben Grossmann Nov 17 '16 at 14:32
  • Um did you mean to swap "former" and "latter"? I can intuitively see that non-self-intersecting homotopy implies a homeomorphism, since we can first prove that the transform can be decomposed into finitely many Reidemeister moves, each of which corresponds to a homeomorphism. It is the other direction I cannot see, probably because I don't know much topology. – user21820 Nov 18 '16 at 04:11
  • @user21820 you're right, I mixed them up. – Ben Grossmann Nov 19 '16 at 19:51
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Knots are embeddings of the circle $\mathbb S^1$ into $\mathbb R^3$. The basic question is, when can one knot be continuously deformed into another?

Robert Israel
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  • Thanks. What are the main topics in knot theory? Could you provide me an example of a typical problem in knot theory, so that I can really understand what knot theory as a branch of study is really concerned about? – user3460322 Nov 16 '16 at 02:12
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    @user3460322: Hanken's Algorithm can distinguish knots but it is really slow. This leads to the study of invariants of knots. An invariant, roughly speaking, is some property that is preserved amongst equivalent knots. (Hopefully an invariant is easy to compute!) How "sensitive" is such an invariant? For example, the Jones Polynomial is an invariant. Every equivalent knot has the same JP. There are some inequivalent knots that possess the same JP. It is unknown whether the JP can distinguish every knot from the circle. – Dair Nov 16 '16 at 02:36
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    Continuously deformed... without passing through itself – Ben Grossmann Nov 16 '16 at 02:44
  • Why R^3? A good answer should address the fact that obstructions do not exist in R^4 and higher dimensions. – Michaël Le Barbier Nov 16 '16 at 06:53
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    @MichaelGrünewald Higher-dimensional knots are interesting too, it just so happens that $\operatorname{Emb}(S^1, \mathbb{R}^d)$ is path-connected for $d > 3$ (but not contractible, which is the interesting part). – Najib Idrissi Nov 16 '16 at 08:42
  • @NajibIdrissi Thank you for this interesting complement! Still I think the answer should hint at why the theory focuses on R^3. – Michaël Le Barbier Nov 16 '16 at 09:44
  • Note that every tame knot is actually continuously isotopic, but not in a way that extends to an isotopy of a neighborhood of the knot. –  Nov 17 '16 at 02:55
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EDIT: This post is not really correct (see comments), but I'll let it stay because I still think there's some value!

When would you say, intuitively, that two knots are the same knot? Your intuitive idea probably coincides strongly with them being homeomorphic topological spaces. (i.e. stretching or bending a knot without cutting or gluing does not change its fundamental knot-soul.) It's important to note that by "knot" what is meant is a loop tangled in some way. Check this examples from this image I took from wikipedia. enter image description here

All of these knots are $\textit{fundamentally}$ different in the sense that to transform one of them into another one of them you must cut and glue the string (which is the idea "topological homeomorphisms" try to capture.)

Some easy questions are already interesting. For example, are there infinitely many different knots?

JKEG
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    All knots are homeomorphic to each other. The issue is finding a homeomorphism of the ambient space that sends one to the other. – Matt Samuel Nov 16 '16 at 02:25
  • @MattSamuel You're right I thought of that just after posting but I couldn't think of how to correctly rewrite my answer and keep it simple. I probably should just delete. – JKEG Nov 16 '16 at 02:31
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    @Mingus Don't delete! – littleO Nov 16 '16 at 02:47
  • And are there infinitely many different knots? Is it known? – Pedro A Nov 17 '16 at 12:24
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    @Hamsteriffic My (naive) intuition is yes, there are an infinite number. If you look at the progression of $3_1$, $5_1$, $7_1$ in the figures above, these go around the "circle" a second time with an increasing number of wraps. Another construction (obvious in $6_1$ and $7_2$) is to hold a loop by the ends, add some number of twists to the loop, and then bring the ends together, and cut and glue the one end around the other. It seems that both constructions can be performed an increasing number of times, but perhaps proving this is hard. Is there an intuitive dis-proof? – toddkaufmann Nov 18 '16 at 00:00
  • This should at the minimum be edited. – Andres Mejia Jul 16 '18 at 01:00
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    @PedroA The answer is yes. You can form the [knot sum](https://en.wikipedia.org/wiki/Connected_sum#Connected_sum_of_knots) of a knot with itself an arbitrary number of times to get infinitely many knots. If you want _prime_ knots, the torus knots form an infinite family. – Akiva Weinberger Nov 22 '18 at 19:51
  • @toddkaufmann I'll ping you too; see my previous comment. As another example, the [Tait conjectures](https://en.m.wikipedia.org/wiki/Tait_conjectures) (now theorems) make it very easy to see when two [alternating knots](https://en.wikipedia.org/wiki/Alternating_knot) are the same. In particular, two ("reduced") alternating knots with different numbers of crossings are different, so to get lots of different knots you can take alternating knots with lots of different amounts of crossings. – Akiva Weinberger Nov 22 '18 at 19:58
  • That's a bit of a worse example, though, since the Tait conjectures were open for over a hundred years, while the torus knots were known to be distinct from the start of knot theory. (I can't find an exact date for when this was proven.) Also: all of the knots in the picture are alternating, because the smallest nonalternating knots have 8 crossings. One of these is the torus knot T(3,4). It is conjectured that most knots are nonalternating. – Akiva Weinberger Nov 22 '18 at 20:05
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As was mentioned numerous times in the other answers and comments, to knot theory is usually studied through the lens of knot invariants. So to give an idea of what invariants are about, I will give some examples of what they can do and can't do.

  1. Being able to compute an invariant is useful for telling knots apart, but not always good for being sure you have the same knot. The easiest to explain is tricolorablity. This is a simple yes or no invariant. A knot is either tricolorable or not. So, if you have two knot diagrams, and one is tricolorable and the either is not, they definitely are not the same knot. But if they are both tricolorable, then we can't tell if they are the same knot or not. So we often are looking for specific classes of knots we can tell apart based on a certain invariant.
  2. One problem with the first item is how easy it is to compute the invariant. Some are relatively easy, but they usually not as useful for telling knots apart. The ones that are are "easiest" (as in, there is an algorithm) to compute are the polynomials - Alexander, Jones, HOMFLY - and they are rather good at distinguishing knots. But the bigger the knots get, the harder the invariants are to compute. To be a little more precise, they all grow exponentially in the number of steps required based on number of crossings in a diagram. (Of course, for a big knot, you might have no chance at computing any invariant, so they still might be the best option.)
  3. Invariants are interesting to see how they act under changes in the knots. For example, we can make new knots from old by a process called connect sum: We just combine two knots into one, by breaking them apart and connecting them into one continuous loop. The bridge number of a knot $b(K)$ is an invariant which gives a number: The bigger the number the more "complex" the knot is. But when we connect sum a knot $J\#K$, we get the nice relation $b(J\#K)=b(J)+b(K)-1$. We think of connect sum as addition of knots, so knot theorists find it pretty and exciting when we almost add in the invariants when we "add" the knots.
  4. We often look at all the knots of a certain value for an invariant. Example: we can look at all knots which are bridge number 2, since I mentioned this invariant already. These knots often have exploitable characteristics which can let us say something about another invariant. Again, keeping with our 2-bridge knots, every two bridge knot has a knot group (another algebraic topology invariant) $\pi_1(S^3-K)$, has a presentation with two generators and one relation.

Obviously, this is just how I look at knot theory, and I am sure you will get different opinions about it from other knot theorists, but I hope this might give a flavor for what it is we do.

N. Owad
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    The online "Knot Exhibition" at http://www.groupoids.org.uk/popmath/cpm/exhib/knotexhib.html was designed, originally as posters in 1989, to use knots to present methods of mathematics to the general public. – Ronnie Brown Nov 16 '16 at 10:23
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We can study knots algebraically also because Quandles are complete knot invariant and quandles are algebraic object. For more information see this http://aleph0.clarku.edu/~djoyce/quandles/aaatswatkt.pdf

Manpreet Singh
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