8a. Knots and links
Leaving the graphs and related topological spaces aside, let us focus now on the simplest objects of topology, which are the knots. Knots are something very familiar, from the real life, and mathematically, it is most convenient to define them as follows:
A knot is a smooth closed curve in [math]\mathbb R^3[/math],
Observe that our knots are by definition oriented. The reverse knot [math]z\to\gamma(z^{-1})[/math] can be isomorphic or not to the original knot [math]z\to\gamma(z)[/math], and we will discuss this in a moment. At the level of examples, we first have the unknot, represented as follows:
For typographical reasons, it is most convenient to represent our knots by squarish diagrams, with these being far easier to type in Latex, the computer program used for writing math books, with the unknot for instance being represented as follows:
The unknot is already a quite interesting mathematical object, suggesting a lot of exciting mathematical questions, for the most quite difficult, as follows:
\begin{questions}
In relation with the unknot:
- Given a closed curve in [math]\mathbb R^3[/math], say given via its algebraic equations, can we decide if it is tied or not?
- Perhaps simpler, given the 2D picture of a knot, can we decide if the knot is tied or not?
- Experience with cables and ropes shows that a random closed curve is usually tied. But, can we really prove this?
\end{questions} Obviously, difficult questions, and as you can see, knot theory is not an easy thing. But do not worry, we will manage to find our way through this jungle, and even come up with some mathematics for it. Going ahead now with examples, as the simplest possible true knot, meaning tied knot, we have the trefoil knot, which looks as follows:
We also have the opposite trefoil knot, obtained by reversing the orientation, whose picture is identical to that of the trefoil knot, save for the orientation of the arrow:
As before with the unknot, while the trefoil knot might look quite trivial, when it comes to formal mathematics regarding it, we are quickly led into delicate questions. Let us formulate a few intuitive observations about it, as follows:
\begin{fact}
In relation with the trefoil knot:
- This knot is indeed tied, that is, not isomorphic to the unknot.
- The trefoil knot and its opposite knot are not isomorphic.
\end{fact} To be more precise, here (1) is something which definitely holds, as we know it from real life, but if looking for a formal proof for this, based on Definition 8.1, we will certainly run into troubles. As for (2), here again we are looking for troubles, because when playing with two trefoil knots, made from rope, with opposite arrows marked on them, we certainly see that our two beasts are not identical, but go find a formal proof for that.
In short, as before with the unknot, modesty. For the moment, let us keep exploring the subject, by recording as Questions and Facts things that we see and feel, but cannot prove yet, mathematically, based on Definition 8.1 alone, due to a lack of tools.
Getting back now to Definition 8.1, as stated, it is convenient to allow, in relation with certain mathematical questions, links in our discussion:
A link is a collection of disjoint knots in [math]\mathbb R^3[/math], taken as usual oriented, and regarded as usual up to isotopy.
As before with the knots, which can be truly knotted or not, there is a discussion here with respect to the links, which can be truly linked or not, and with orientation involved too. Drawing some pictures here, with some basic examples, is very instructive, the idea being that two or several basic unknots can be linked in many possible ways. For instance, as simplest non-trivial link, made of two unknots, which are indeed linked, we have:
By reversing the orientation of one unknot, we have as well the following link:
This was for the story of two linked unknots, which is easily seen to stop here, with the above two links, but when trying to link [math]N[/math] unknots, with [math]N=3,4,5,\ldots\,[/math], many things can happen. Which leads us into the following philosophical question:
\begin{question}
Mathematically speaking, which are simpler to enumerate,
- Usual knots, that is, links with one component,
- Or links with several components, all being unknots,
and this, in order to have some business started, for the links? \end{question} And with this being probably enough, as preliminary experimental work, time now to draw some conclusions. Obviously, what we have so far, namely Questions 8.2, Fact 8.3 and Question 8.5, is extremely interesting, at the core of everything that can be called “geometry”. And by further thinking a bit, at how knots and links can be tied, in so many fascinating ways, we are led to the following philosophical conclusion: \begin{conclusion} Knots and links are to geometry and topology what prime numbers are to number theory. \end{conclusion} Very nice all this, we are now certainly motivated for studying the knots and links, and time for some mathematics. But the question is, how to get started?
In view of the above, this is not an easy question. Fortunately, graph theory comes to the rescue, via to the following simple fact, which will be our starting point:
\begin{fact} The plane projection of a knot or link is something similar to an oriented graph with [math]4[/math]-valent vertices, except for the fact that we have some extra data at the vertices, telling us, about the [math]2[/math] strings crossing there, which goes up and which goes down. \end{fact} Based on this, let us try to construct some knot invariants. A natural idea is that of defining the invariant on the 2D picture of the knot, that is, on a plane projection of the knot, and then proving that the invariant is indeed independent on the chosen plane.
This method rests on the following technical result, which is well-known:
Two pictures correspond to plane projections of the same knot or link precisely when they differ by a sequence of Reidemeister moves, namely:
- Moves of type I, given by [math]\propto\ \leftrightarrow\,|[/math].
- Moves of type II, given by [math])\hskip-1.7mm(\ \leftrightarrow\ )([/math].
- Moves of type III, given by [math]\vartriangle\ \leftrightarrow\,\triangledown[/math].
This is something very standard, as follows:
(1) To start with, the Reidemeister moves of type I are by definition as follows:
(2) Regarding the Reidemeister moves of type II, these are by definition as follows:
(3) As for the Reidemeister moves of type III, these are by definition as follows:
(4) This was for the precise statement of the theorem, and in what regards now the proof, this is somewhat clear from definitions, and in practice, this can be done by some sort of cut and paste procedure, or recurrence if you prefer, easy exercise for you.
At a more advanced level now, we will need the following key observation, making the connection with group theory, and algebra in general, due to Alexander:
Any knot or link can be thought of as being the closure of a braid,
Again, this is something quite self-explanatory, as follows:
(1) Consider indeed the braids with [math]k[/math] strings, with the convention that things go from up to down. For instance the braid in the statement should be thought of as being:
But, with this convention, braids become some sort of permutations of [math]\{1,\ldots,k\}[/math], which are decorated at the level of crossings, with for instance the above braid corresponding to the following permutation of [math]\{1,2,3,4,5\}[/math], with due decorations:
In any case, we can see in this picture that [math]B_k[/math] is indeed a group, with composition law similar to that of the permutations in [math]S_k[/math], that is, going from up to down.
(2) Moreover, we can also see in this picture that we have a surjective group morphism [math]B_k\to S_k[/math], obtained by forgetting the decorations, at the level of crossings. For instance the braid pictured above is mapped in this way to the following permutation in [math]S_5[/math]:
It is possible to do some more algebra here, in relation with the morphism [math]B_k\to S_k[/math], but we will not need this in what follows. We will keep in mind, from the above, the fact that “braids are not exactly permutations, but they compose like permutations”.
(3) Regarding now the closure operation in the statement, this consists by definition in adding semicircles at right, which makes our braid into a certain oriented link. As an illustration, the closure of the braid pictured above is the following link:
(4) This was for the precise statement of the theorem, and in what regards now the proof, this can be done by some sort of cut and paste procedure, or recurrence if you prefer. As before with Theorem 8.8, we will leave this as an easy exercise for you.
Many interesting things can be said about the braid group [math]B_k[/math], as for instance:
The braid group [math]B_k[/math] has the following properties:
- It is generated by variables [math]g_1,\ldots,g_{k-1}[/math], with the following relations:
[[math]] g_ig_{i+1}g_i=g_{i+1}g_ig_{i+1}\quad,\quad g_ig_j=g_jg_i\ {\rm for}\ |i-j|\geq2 [[/math]]
- It is the homotopy group of [math]X=(\mathbb C^k-\Delta)/S_k[/math], with [math]\Delta\subset\mathbb C^k[/math] standing for the points [math]z[/math] satisfying [math]z_i=z_j[/math] for some [math]i\neq j[/math].
These are things that we will not really need here, as follows:
(1) In order to prove this assertion, due to Artin, consider the following braids:
We have then [math]g_ig_j=g_jg_i[/math], for [math]|i-j|\geq2[/math]. As for the relation [math]g_ig_{i+1}g_i=g_{i+1}g_ig_{i+1}[/math], by translation it is enough to check this at [math]i=1[/math]. And here, we first have:
On the other hand, we have as well the following computation:
Now since the above two pictures are identical, up to isotopy, we have [math]g_1g_2g_1=g_2g_1g_2[/math], as desired. Thus, the braid group [math]B_k[/math] is indeed generated by elements [math]g_1,\ldots,g_{k-1}[/math] with the relations in the statement, and in what regards now the proof of universality, this can only be something quite routine, and we will leave this as an instructive exercise.
(2) This is something quite self-explanatory, based on the general homotopy group material from chapter 7, and we will leave this as an easy exercise for you.
(3) Finally, before leaving the subject, let us mention that the Artin relations in (1) are something very useful, in order to construct explicit matrix representations of [math]B_k[/math]. For instance, it can be shown that the braid group [math]B_k[/math] is linear, and well, we will leave this as usual as an exercise for you, meaning either solve it, or look it up.
General references
Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].