Knot Theory

Playing with strings

Knot theory is one of those topics where you start out by asking a very simple and natural question, follow a thread (hehe), then look around you and realize you’re knee-deep in at least 5 fields of math.

The central topic of interest within knot theory is - you guessed it - knots. A knot in this context can be thought of as just a piece of string that is attached together at the ends. So if you feel like it, go and grab a piece of string lying around your house, or cut open a rubber band or whatever. It’s literally all you need.

Disclaimer: I’m not going to write down any proofs in here. I’ll just mention that it is proven and then you can go down to the lecture series videos if you are interested in them. (Good proofs often also illuminate the why, so if a jump in logic here doesn’t make sense be sure to look it up.) Also, I’m just learning to work with Inkscape to draw graphics whenever I can’t find them on Wikipedia, so cut me some slack in that department.

The simplest possible knot we can look at is made when we take our piece of string and just tie the ends together so that it makes a clean circle. This is such a simple knot it is barely even a knot, right? This is why it’s called the un-knot. It’s depicted on the left in the figure below:

Fig: The unknot in 2 different forms. Note that we are taking an object existing in 3D space and drawing a projection of it in a 2D image: The line being invisible for a little bit denotes the fact that it crosses underneath itself there

But wait a minute - what about the knot on the right? If we would grab that string by the top bit and rotate that around, we would get the unknot again. Because this is possible, we say that they are both the same knot. We are not really interested in those differences that can be removed through actions like these (called ambient isotopies). This means that the unknot can take very complicated forms.

But how do we know if a knot is the unknot when all we see is a complicated diagram? Should we simply get a piece of string and try to pull from all directions until we get the unknot back? What if we don’t succeed? Does that mean it’s not actually the unknot, or that we just didn’t try for long enough?

It turns out that there exist many different knots. There is a system of enumerating them, by considering each knot in their simplest form: one aspect of this means having the minimum amount of crossings (the times that the diagram crosses underneath itself). If we call this minimum crossing number $C$, a knot is referred to as $C_i$, where $i$ is simply a number allocated to it in a knot-encyclopedia of sorts, but not inherently meaningful.

Fig: The table of prime knots up to 7 crossings with Alexander-Briggs notation (Source)

The simplest form of a knot is called a prime knot. Yes, that word carries a lot of baggage with it, and no it doesn’t just talk the talk. Prime knots are not the result of composing together other simpler knots, and all knots can be uniquely constructed by connecting prime knots together. (Composing a knot is as simple as cutting both of them open, and connecting the ends of the knots together.) Prime knots are the equivalent of prime numbers within number theory. The above table can still be depicted very comfortably here, but the number of prime knots as a function of crossing number increases dramatically quick.

What I didn’t say yet is how we know we know that these are all different knots, and how we know that these are all knots with up to 7 crossings. This is a really hard question. Finding out if two knots are equivalent to eachother is actually the central question of knot theory, and it leads to some interesting places.

De-composing movements

As it turns out, there are actually 3 types of ways you can deform a knot to make it change its appearance, without altering what knot it is. These ways are called the Reidemeister moves. Each one looks at a section of the knot, and what you can do with it:
Fig: The 3 Reidemeister moves

The moves are as follows:

1. Taking a part of the knot, and twisting a bit of it into a simple new loop crossing underneath itself
2. Having two separate parts of the knot, and pulling a small bit of the one underneath the other (pulling blue underneath red in the image)
3. Having two sections crossing over eachother (blue and green in the image), and a third section passing over each of them (red), you can move that third section completely over the initial crossing (move red downwards)

This means that any two diagrams of the same knots can be transformed into eachother through a sequence of the correct Reidemeister moves. You probably see quickly for the two depictions of the un-knot pictured above that it is just one step: move 1. However, for larger knots with many crossings, even this quickly becomes unviable. Consider also the knots $3_1$ and $4_1$ in the diagram above: Knowing the Reidemeister moves doesn’t immediately show you that it is impossible to transform one into the other through them. Perhaps there is an extremely long sequence of moves that takes you from one to the other. You could check all possible sequences of a certain length of Reidemeister moves, but the sequence length you should check increases absolutely monstrously quick as a function of crossing number. Proving the negative in a viable manner requires an extra ingredient: invariants.

Knot Invariants

Much of knot theory consists of mapping a knot to some number, which is not dependent on the way that it is folded. To prove that the quantity is invariant to all possible transformations, we can still use the Reidemeister moves. If we prove that the number stays constant under application of each of the 3 Reidemeister moves, we know it’s an invariant.

Colorability

As a simple example of this, we can define tri-colorability. This is a binary outcome which says if the knot $K$ is tri-colorable or not. Taking a 2D knot diagram, we should try to color each line segment (so stopping when the line goes under a crossing) according to the following rules:

1. At least 2 different colors have to be used
2. At each crossing, the lines have to be either all the same color, or all different

If this is possible, the knot is called tri-colorable. An example is given below:

Fig: A Granny Knot which has been tri-colored (source)

The Reidemeister moves don’t affect this binary tri-colorability invariant. So now we know that if one knot is tri-colorable while the other is not, they must be different knots. To flex our newfound knowledge, let’s refer back to the table of prime knots and try to find the tri-colorability of both the un-knot and $3_1$. The un-knot is not tri-colorable, as we fail immediately at rule #1. The knot $3_1$ is the simplest knot to tri-color (try it!). This means we’ve proven that the first 2 knots in the diagram are different! Unfortunately this invariant is rather weak: As it’s a binary property, it can only ever divide all possible knots into two groups, without any way to discern between knots inside each group. The granny knot pictured above is also tri-colorable, but we know it’s not $3_1$.

The solution is clear: Construct more invariants (preferably more discerning than this one) and start throwing them at our collection of knots to find if they can discern between different ones.

Alexander Polynomial

One of the invariants derived by James Alexander in 1923 is the Alexander polynomial. As you may expect, producing a polynomial for each knot should be much more distinctive in theory than a binary outcome could ever be.

The following is a recipe for taking a knot diagram and producing the Alexander polynomial from it:

1. Orient the knot (this means assigning a consistent direction in which you can travel along your piece of string). You now have a diagram with $n$ crossings and $n+2$ regions bounded by line segments. (To see why, you can consider the un-knot: 0 crossings, but there are 2 regions: the area inside it and the one outside it. For every time that you twist it, one crossing is added, and another area inside is added.)
2. Now build an $n$-by-($n+2$) incidence matrix with an entry for each crossing/region combination (a row for each crossing, a column for each region)
3. For each crossing (refer to figure below):
   • Align yourself to the piece of the knot coming in to and traveling underneath the crossing
   • The region to the left, before the crossing gets assigned a $-t$, the one on the left after crossing a $t$.
   • The region to the right before crossing gets assigned a 1, after crossing a -1
   • If a region doesnt touch this crossing at all, assign it a 0
4. Take your completed incidence matrix and delete any 2 columns of your choosing corresponding to adjacent regions
5. Calculate the determinant of this matrix. You will end up with a polynomial in t
6. Divide out the largest power of t^n and multiply by 1 if needed to make the constant term positive. (This is needed to remove ambiguity in the polynomial due to your freedom in choice of columns to delete during step 4.)

Fig: Constructing the incidence matrix from each crossing for the Alexander polynomial

Example

Now we’ll run through how you can do this for the knot $4_1$. In the figure below we have already picked an orientation for the knot, and have numbered both the crossings and regions. Note that all these choices are arbitrary and will not affect the final polynomial.

The knot $4_1$ with an orientation, crossings (in blue) and regions (in green) numbered to build an incidence matrix

We go through the crossings one by one, starting at the one you first encounter while moving upwards on the left side. This results in the following incidence matrix (If you want, verify for a crossing or two that the numbers indeed end up where you expect them to):

$$ \begin{bmatrix} -t & 1 & t & -1 & 0 & 0 \newline -1 & 0 & t & -t & 1 & 0 \newline -t & t & 0 & 0 & 1 & -1 \newline 0 & 1 & 0 & -t & t & -1 \newline \end{bmatrix} $$

Now throwing away the first two columns and calculating the determinant of the remaining 4x4 matrix we get $-t^3 + 3t^2 - t$. This we can then divide by $t^2$ (no need to multiply by $-1$ here) to arrive at the Alexander polynomial of $4_1$:

$$ \Delta_{4_1}(t) = -t - t^{-1} + 3 $$

To check our work we can go to this Knot Atlas Wiki page, click on the knot $4_1$ and look at the invariants section.

Derivation/meaning of the Alexander polynomial

This whole procedure is a bit magical when you see it, but remember: as long as you can prove none of the Reidemeister moves would change the outcome, you are good.

The actual derivation is based on the fundamental group, a concept in algebraic topology which is really cool in its own right. Bit heavy on the theory, but it comes down to studying how the knot $K$ subdivides the $\Reals^3$ space itself into pieces where a loop of string would get stuck if you try to move it around from one place to the other.
For example: I can take a ring from my finger (an unknot), and then study how it subdivides $\Reals^3$: Get a couple of string pieces, and see how many different simple loops you can make that cannot be transformed into eachother. See the picture below for illustration. Together with the operation of adding two loops together, it turns out that this space of different loops is actually a group isomorphic to $\Z$.

Fig: Each of the three simple/unknotted red oriented loops cannot be deformed into eachother without ‘breaking through’ the orange ring. For this set up if loops are wrapped around the central ring the same number of times, they can be deformed into eachother.

I hope it makes sense to you that this is very close to what knots are at their core: a piece of string restricting its free movement through space by the fact that a part of it is blocking itself. It then is a small step to see that the properties of this division of the space, $\Reals^3 \setminus K$, is a knot-invariant. You could not wiggle the red loops around the ring free in the simple example above, and neither will deforming the ring itself help. This fact is then exploited through some snazzy group theory to end up with the recipe for the Alexander polynomial described above.

The Alexander polynomial is very effective at telling knots apart: It is different for all prime knots up to crossing number 8! However, it cannot discern between a knot and its mirror image. But of course people have been at it for about 100 years by now, so a lot more invariants have been found with differing properties.

Other invariants

• Conway polynomial: A recursively-defined polynomial in $z$ with integer values for the coefficients, commonly written $\nabla(z)$. It defines the un-knot to have a polynomial equal to $1$, together with a recursive relationship: $\nabla(L_{+}) - \nabla(L_{-}) = z\nabla(L_{0})$, where $L_{+}$ and $L_{-}$ can be ‘made’ from $L_{0}$ by taking two separate parts of a knot, and connecting the strings together in 2 different ways (see the figure below). In this way more and more complicated knots can be constructed from the un-knot.
• Genus: This approach considers the lines of the knot to bound a certain surface (called the Seifert surface). The genus of this surface (the number of holes in it) is an invariant.
• Chirality: If you take the mirror image of a knot, will you get the same knot? If you get a different one, that knot is chiral
• Jones polynomial: Like the Conway polynomial, it’s recursively defined by having the value 1 for the un-knot, but a slightly different recursion: $(t^{\frac{1}{2}} - t^{-\frac{1}{2}})V(L_{0}) = t^{-1}V(L_{+}) - tV(L_{-})$. Unlike the Alexander polynomial, the Jones polynomial can in some cases discern between a chiral knot and its mirror image as well.
• Concordance/slice: Embedding the knot in a 4D hypersphere. If this can be done in a locally flat/smooth way, the knot is said to be slice.

Fig: How to perform knot surgery for the purposes of the Conway polynomial (source)

There are more invariants like this as you may have seen listed in the Knot Atlas wiki link. Some are generally stronger than others, but none of them are found to be distinctive enough to discern between all different knots. For example, it is still an open problem whether there exist other knots than the un-knot with Jones polynomial $V(K) = 1$. When trying to solve the question of whether two given knots are the same oftentimes multiple of these invariants are computed and compared against eachother to obtain a higher degree of confidence that they are the same. But only finding the correct series of Reidemeister moves can give you certainty.

Further reading

Hopefully this brief intro to knot theory has shown you how very simple questions around strings can lead to interesting approaches in all kinds of fields: (algebraic) topology, group theory, linear algebra, polynomials, etc.
At this point we have just tried to answer the most obvious question, but did not for example look at the linking together of multiple knots, braiding, what happens when we consider torsion in the string, or that they have a finite thickness as well. Nor did we look at applications in biology, physics or taking your knitting game to the next level. Each of these is a whole other rabbit hole.

For some more info take a look at:

• A really great lecture series on knot theory by Dr. Bosman at Andrews University. The Youtube video playlist starts with a nice entry level explanation to the central topic, but goes very in depth and the lecturer is excellent. Most of the stuff on this page is based on this.
• The Rolfsen knot table in the Knot Atlas. Nice reference information and pictures.