[GENTLE MUSIC] When you think about math, what do you think of?
How about knots, as in actual tangles and knots?
There's mathematics behind this too, and mathematicians are using it right now to better understand DNA.
[ELECTRONIC MUSIC] Today I'd like to tell you about a special kind of mathematical tangle called a rational tangle, first defined by mathematician John Conway around 1970.
Later, we'll see how this relates back to biology and the study of DNA.
What's a rational tangle?
Imagine you have two strings, and each end is attached to the inside of a sphere, but the ends aren't glued in place.
They're not fixed, so they can easily slide around like this.
Now, for simplicity, instead of drawing the full sphere, we'll just show a circle as the boundary.
We'll start with the strings in this position, where they don't cross each other.
It's called the zero tangle.
Now, from here, there are two types of twists you can do, and any combination of these twists will tangle up the strings.
The result you get is called a rational tangle because it corresponds to a rational number.
We'll see how in a second.
But what are the two moves?
First, you can twist horizontally.
Imagine holding the left ends of the two strings while twisting the right ends around each other.
Let's say counterclockwise if we're looking this way.
So notice that the top string goes over.
This move corresponds to plus one.
And if you wanted to rotate clockwise so that the top string goes under, then the move corresponds to minus one.
That's the first move.
Now, for the second move, if your strings are positioned like this, then we can twist vertically and pick up a plus one again.
By vertically I mean imagine holding the tops of the strings in place while twisting the bottom ends around each other, say clockwise if we're looking up at it this way.
Notice that the left string goes over.
Again, this corresponds to plus one or minus one if we twist counterclockwise.
Let's call this one a vertical twist and the previous twist a horizontal twist.
Now, using these two moves, we can create tangles which have their own number.
For example, the tangle you get after two horizontal twists corresponds to the number two.
Or if you do three twists, you get three.
Four twists, you get four, and so on.
Similarly, you can do two, three, four more vertical twists too.
But it gets interesting when we alternate or combine the moves-- horizontal twist, vertical twist, horizontal twist, vertical twist.
The number for the resulting tangle comes from a continued fraction.
Here's the rule.
Start with a certain number of horizontal twists, and then alternate vertical, horizontal, and so on.
If the last move you do is a vertical twist, then the continued fraction is zero plus one over the last number in the list plus one over the second to last number in the list plus one over the third to last number in the list and so on.
Otherwise, if the last move you do is a horizontal twist, then the continued fraction is just the last number plus one over the second to last number plus one over the third to last number and so on.
It's way easier than it sounds.
For example, let's first do two horizontal twists followed by three vertical twists.
This corresponds to the continued fraction 0 plus 1 over-- and now look at the last move-- 3 plus 1 over the first move, 2.
The result is 2/7.
But if we tack on four more horizontal twists, this time the fraction is start with the last move, 4, plus 1/3 plus 1/2, which is 30/7.
Then we can add two more vertical twists, and so now the fraction is 0 plus 1/2 plus 1/4 plus 1/3 plus 1/2, which is 30/67.
OK. See the pattern?
A rational tangle is an alternating combination of vertical and horizontal twists, which we can represent by numbers in a list.
For example, two, three, zero means do two horizontal twists, then three vertical twists, and stop, while two, three, four means do two horizontal twists, then three vertical twists, then four horizontal twists.
And these numbers give us a continued fraction.
Now here's the cool thing.
You might know that there's more than one way to write a rational number as a continued fraction.
For example, 30/7 is, as we saw, 4 plus 1/3 plus a half.
But-- and you can check-- it can also be written like this.
Now here's the rational tangle corresponding to that continued fraction, and here's the rational tangle corresponding to the first one.
And because their two fractions are actually the same, you'd really hope that those two tangles are the same too.
And it's true.
They're the same in the sense that you can actually get from one tangle to the other without, number one, tearing the strings or, number two, moving any of the four end points.
The upshot is that every rational tangle corresponds to a rational number.
But even better, the tangle is uniquely determined by its rational number, meaning any two rational tangles whose continued fractions are equal are the same rational tangle.
In particular, the converse to the story also holds.
Pick any rational number, then you can always find a rational tangle and tangles equivalent to it that have that number as its continued fraction.
In other words, there is a bijection between equivalence classes of rational tangles and rational numbers, including infinity.
And this is called the rational tangle theorem.
By the way, not every tangle of two strings is a rational tangle, so an alternating combination of horizontal and vertical twists.
For instance, this one is not.
Can you figure out why?
At this point, there are tons of questions that you can ask, like, hey, we can add and multiply rational numbers, so is there a way to add and multiply rational tangles?
Can you start from any tangle and get back to the zero tangle or conversely?
And, oh, by the way, why are continued fractions the right thing to think about in the first place?
There are lots of things to explore.
We're not even scratching the surface.
If you want to dig deeper-- and I hope that you do-- I put a ton of great links below.
But I want to quickly mention that rational tangles show up in the real world too, and I mean that literally, inside your body.
In particular, in your DNA.
Now, having knots in your DNA is in general not good in the following sense.
When your body creates new cells, your DNA replicates, and during this process of cell division, the DNA helix splits down the middle.
That way, each of the two strands can serve as a template for new DNA.
But notice what happens if they're knotted or linked together.
That's bad because the two pieces of DNA can't be separated to create individual new cells.
So what you really wish you could do is go in there with tiny scissors, cut the link, pass one piece of DNA through the other and then reseal.
That way, the two pieces become unlinked and are free to be replicated.
OK. Not only is that possible, it happens all the time naturally right now in your body, except the tiny scissors are an enzyme called topoisomerase.
So bottom line, having knots in your DNA can cause your cells to die, which is bad, and so topoisomerase is good unless that cell is bad to begin with, like a bacterial or a cancer cell.
In that case, you do not want topoisomerase to help the DNA duplicate, so drugs have been manufactured that will block that specific enzyme.
But the problem is that it's hard to differentiate between a normal cell and a cancer cell.
So if you block topoisomerase from working on a cancer cell, it'll stop working on a healthy cell too.
How do you fix this?
Well, it's an active area of research, and rational tangles can be used to model and study the enzyme's behavior, because the better we understand its behavior, the better we can do things like tell it to assist healthy cells and avoid cancer cells.
And speaking of current events, rational tangles were recently featured on a great math podcast called "My Favorite Theorem."
I've put a link to the website below.
Take a look at episode 12, where you can listen to mathematician Candice Price talk more about the rational tangle theorem and this area of math called DNA topology.
By the way, today's discussion falls under the larger subject of knot theory, the mathematical study of knots.
Knots are of interest both to applied and to pure mathematicians.
Knot theory also shows up in quantum physics, in string theory, and in something called a topological quantum field theory.
Think about that next time you look at your headphones.