Inside a classroom divide students into groups such that each group has at least 5 students. Groups should stand up and form a circle before following the upcoming instructions:
1. If there is an even number of student in a group:
Have each student extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
Repeat the same thing for the left hand.
2. If there is an odd number of student in a group:
Have each student (except one of them) extend his/her right hand and take the right hand of another student in the group who is not adjacent to him/her.
Then take the extra student and have him/her extend his/her right hand so that the extra student can hold the left hand of another student who is not adjacent to him/her.
Finally, repeat the process for the students whose left hands are free.
In the end, students will be knotted.
Now, each group should find a way to untangle themselves without letting their hands go. To do that, one can use Reidemeister moves.
Back in 1926 Kurt Reidemeister discovered something very useful in knot theory. According to him, in the knot theory one can use three moves which we call after his name. Using these three moves we can show if two (or more) knots are the same or not.
For example, using Reidemeister moves, we can see if a knot is an unknot (in other words, if it can be untangled or not).
What are these moves?
First Reidemeister move is twist. We can twist (or untwist) a part of a knot within the knot theory rules.
Second one is poke. We can poke a part of a knot as long as we don’t break (or cut) the knot.
Final one is slide. We can slide a part of a knot according to Reidemeister.
After you participating in a human knot game, ask yourselves which Reidemeister moves did you use during the game?
In my youth, I would never step outside without my cassette player. Though, I had two knotting problems with my cassette player: the First one was about the cassette itself. Rewinding cassettes was a big issue as sometimes the tape knotted itself. Whenever I was lucky, sticking a pencil would solve the whole problem.
The second knotting problem was about the headphone. Its cable would get knotted so bad, it would take me 10 minutes to untangle it. Most of the time I would bump into a friend of mine and the whole plan about listening to music would go down the drain.
The funny thing is I would get frustrated and chuck the headphone into my bag which would guarantee another frustration for the following day.
A similar tangling thing happens in our body, inside our cells, almost all the time.
DNA: A self-replicating material that is present in nearly all living organisms as the main constituent of chromosomes. It is the carrier of genetic information.
DNA has a spiral curve shape called “helix”. Inside the cell, the DNA spiral sits at almost 2-meters. Let me give another sight so that you can easily picture this length in your mind: If a cell’s nucleus was at the size of a basketball, the length of the DNA spiral would be up to 200 km!
You know what happens when you chuck one-meter long headphones into your bags. Trying to squeeze a 200 km long spiral into a basketball?! Lord; knots everywhere!
This chaos itself was the reason why mathematicians got involved with knots. Although, knot-mathematics relationships existed way before DNA researches started. In the 19th century, a Scottish scientist named William Thomson (a.k.a. Lord Kelvin) suggested that all atoms are shaped like knots. Though soon enough Lord Kelvin’s idea was faulted and knot theory was put aside for nearly 100 years. (At the beginning of the 20th century, Kurt Reidemeister’s work was important in knot theory. I will get back to Reidemeister in the next post.)
Is there a difference between knots and mathematical knots?
For instance the knot we do with our shoelaces is not mathematical. Because both ends of the shoelace are open. Nevertheless a knot is mathematical if only ends are connected.
The left one is a knot, but not mathematically. The one on the right is mathematical though.
Unknot and Trefoil
In the knot theory we call different knots different names with using the number of crosses on the knot. A knot that has zero crossing is called “unknot”, and it looks like a circle:
Check out the knots below:
They look different from each other, don’t they? The one on the left has 1 crossing as the one on the right has 2:
But, we can use manipulations on the knot without cutting (with turning aside and such) and turn one of the knots look exactly like the other one!
This means that these two knots are equivalent to each other. If you take a good look you will see that they are both unknots. For example, if you push the left side of the left knot, you will get an unknot:
Is there a knot that has 1 crossing but can’t be turned into an unknot?
The answer is: No! In fact, there are no such knots with 2 crossiongs either.
How about 3 crossings?
We call knots that have 3 crossings and that can’t be turned into unknots a “trefoil”.
Even though in the first glance you would think that a trefoil could be turned into an unknot, it is in fact impossible to do so. Trefoil is a special knot because (if you don’t count unknot) it has the least number of crossings (3). This is why trefoil is the basis knot for the knot theory.
One of the most important things about trefoils is that their mirror images are different knots. In the picture, knot a and knot b are not equivalent to each other: In other words, they can’t be turned into one another.
Möbius Strip and Trefoil
I talked about Möbius strips and its properties in an old post. Just to summarize what it is; take a strip of paper and tape their ends together. You will get a circle. But if you do it with twisting one of the end 180 degrees, you will get a Möbius band.
Let’s twist one end 3 times:
Then cut from the middle of the strip parallel to its length:
We will get a shape like the following:
After fixing the strip, you can see that it is a trefoil knot:
To be continued…
1. In order to make a trefoil knot out of a paper strip, we twist one end 3 times. Is there a difference between twisting inwards and outwards? Try and observe what you end up with.
2. If you twist the end 5 times instead of 3, what would you get? (Answer is in the next post.)