Perko pair knots

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Perko pair knots

This is a picture of the Perko pair knots. They were first thought to be separate knots, but in 1974 it was proved that they were actually the same knot.


Contents

Basic Description

In 1899, C. N. Little published a table of 43 nonalternating knots of 10 crossings that listed the two knots shown above as being distinct. Seventy-five years later, Kenneth Perko, a lawyer and part-time mathematician, discovered that these were actually the same knot[1].

To say that two knots are the same is to say that one can be deformed into the other without breaking the knot or passing it through itself. To prove that two knots are the same, we can create one of them out of actual rope, and tug at it and move it around until it looks like the other. As the story goes, that's how Perko figured out that these knots are the same - by working with rope on his floor.

We can also prove that two knots are the same by working with their projections. A projection of a knot is a flat representation of it, essentially a 2D drawing of the knot. There are many ways to use projections to show that certain knots are distinct from each other, but the main way of using projections to demonstrate that two knots are the same is to use the Reidemeister moves, which are described below.

A More Mathematical Explanation

Reidemeister moves

As was stated above, knots are considered to be the same if one can be rearra [...]

Reidemeister moves

As was stated above, knots are considered to be the same if one can be rearranged into the other without breaking the string or passing it through itself. This kind of transformation is called an ambient isotopy. But when we're writing a written proof, we have to work with the knots projection, instead of the knot itself. What manipulations can we make on a knot’s projection that correspond to ambient isotopies in three dimensions?


The first answer is a planar isotopy. A planar isotopy is the sort of transformation you could make if the projection of a knot was printed on very stretchy rubber. The image can be stretched in all directions, but none of the crossings are affected:

The original image.

These two images are planar isotopies of the original image.

This is not a planar isotopy of the original image.


The second answer is the Reidemeister moves, a set of three changes we can make to a knot’s projection that do affect the knot’s crossings but are still ambient isotopies. Every change to a knot's projection that corresponds to an ambient isotopy can be described as some combination of these three moves. In the images below, we imagine that the line segments continue and connect in some sort of unspecified knot, and only the section of the knot we're looking at changes:

Type I Reidemeister Move:

The first Reidemeister move allows you to create a twist in a strand that goes in either direction.

Type II Reidemeister move:

The second Reidemeister move allows you to slide one strand on top of or behind another.

Type III Reidemeister move:

The third Reidemeister move allows you to slide a strand to the other side of a crossing.


Proving that the Perko knots are equivalent

In his paper "On the Classification of Knots", Kenneth Perko provided an abridged proof that the knots now known as the Perko pair are the same[2]. This proof is shown below:


Perko's proof relies on the ability of the reader to manipulate the knots in their head and verify that each projection can be manipulated to look like the next. To create a full, rigorous proof, we need to use planar isotopies and the Reidmeister moves, as described above.

Below is a step-by-step Reidemeister moves proof that follows the outline of Perko's shorter proof. The arrows between each step are labeled to show how we get from one image to the other: p.i. means we use a planar isotopy, I means we use the first Reidemeister move, II means we use the second move, and III means we use the third move. Mousing over a step will highlight the part of the knot that's about to move in pink, and display a dotted green line showing where it will move to.



Dowker notation

Dowker notation is a way of describing knots with numbers. Each crossing is labeled with two numbers, and anyone who knows the notation can reconstruct the knot by connecting the numbers in the right order[1].

Image 1. The first Perko knot with direction assigned.
Image 1. The first Perko knot with direction assigned.
Image 2. The first Perko knot with numbers assigned to the crossings. Orange numbers are assigned when going over a crossing, pink numbers are assigned when going under.
Image 2. The first Perko knot with numbers assigned to the crossings. Orange numbers are assigned when going over a crossing, pink numbers are assigned when going under.

To determine a knot's Dowker representation, first we need to assign direction to the knot. This is shown in Image 1.


Next, pick any crossing, and assign the number 1 to it. Follow the strand that goes under that crossing to the next crossing, and label it 2. Make sure to travel in the direction specified by the arrows. Keep following the arrows around the knot, and assigning numbers sequentially to every crossing. When going under a crossing, negative even numbers are assigned in place of positive even numbers[3]. In Image 2, we can see that the second number is -2 instead of 2, because it was assigned while going under. All odd numbers are positive.


We continue all the way around the knot until every crossing has two numbers, one for the strand that goes under and one for the strand that goes over as shown in Image 2. At each crossing, one of the numbers is odd and the other is even.


Then we put our numbers into a table. On the top row, we put the odd numbers in numerical order. Beneath each odd number, we put the even number that is found for the same crossing. The table that corresponds with Image 2 is shown below:

1

3

5

7

9

11

13

15

17

19

14

-10

16

-18

-2

-6

20

4

-12

-8

The list of even numbers on the bottom row of the table is the knot's Dowker representation.


Image 3. The beginning of the number line used to create a picture of a knot from its Dowker representation.
Image 3. The beginning of the number line used to create a picture of a knot from its Dowker representation.

To get from Dowker notation to a picture of a knot, we begin by laying out a number line. At each tick mark on the number line, we write both the number that would normally be at that spot and the number that it's matched with in the Dowker representation of the knot[1]. This is shown in Image 3, where the points on the number line labeled 1, -2, and 3 are also labeled 14, 9, and 10.


We draw the number line until we reach a number that is already labeled on it. In our example, this happens when we reach 9, which is already on the number line as the counterpart of -2. Since the number is already on the number line, we circle our line back around to go through the crossing at -2 and 9. This is shown in Image 4 below:


Image 4. Since 9 is already on the number line, we circle back around and cross the number line.
Image 4. Since 9 is already on the number line, we circle back around and cross the number line.


We continue in this manner for the rest of the number pairs in our list. If the number is already on the drawing, we circle around to connect to it. If not, we draw a tick mark on our line to represent a crossing that we will come back to later. Once we reach the last number, which is 20 in our example, we connect the line up with the beginning of the number line. We then go back through the knot and use the negative signs to assign crossings.

Image 5. The final image. Blue numbers are numbers reached before leaving the number line, green numbers are those reached after leaving the line.
Image 5. The final image. Blue numbers are numbers reached before leaving the number line, green numbers are those reached after leaving the line.
Image 6. Another projection of the Perko knot, with a different Dowker representation.
Image 6. Another projection of the Perko knot, with a different Dowker representation.


Image 5 is the knot we get when we apply this process to 14, 10, 16, -18, -1, -6, 20, 4, -12, -8 (the Dowker representation of the first Perko knot). We can get from this image to our original image using only planar isotopies.


Dowker representations are unique to a specific projection of a knot, not to the knot itself. One set of Dowker numbers will always generate the same projection of a knot, but there may be other projections of that same knot that have a different Dowker representation.


One easy way to see this is to remember that we can always create a projection of a knot that has more than the minimum number of crossings. The Perko knot is a ten crossing knot, which means there is no projection of it with fewer than ten crossings, but in our proof above, our intermediate steps generally had eleven or more crossings. If the number of crossings is different, it follows that the Dowker representation must be different too, because we will need a different number of labels.


Projections with the same number of crossings can have different Dowker representations as well. For example, if we look at the second Perko pair knot, we can see in Image 6 and in the table below that it has a different Dowker representation than the first knot.

1

3

5

7

9

11

13

15

17

19

16

-10

14

-18

-4

-20

6

2

-12

-8

It might seem that being able to represent a knot in more than one way is a shortcoming of Dowker notation, but it's actually quite useful to have a tool that is specific to the projection being worked with. As the case of the Perko knots shows, different projections of a knot can look like totally different knots, so it's valuable to have a way to make sure that everyone is looking at the same projections.



Why It's Interesting

The story of the Perko knots really helps to show that mathematics, and knot theory in particular, is not limited to academics and professional mathematicians. Anyone who is interested in math and spends time studying it has the potential to discover something new and interesting, no matter what their day job is.

Knot theory itself is one of the most accessible areas of math. What could be more familiar than knots? We use them every time we put on a pair of sneakers or a tie, and they're an important part of many crafts, from knitting to embroidery. Knots are also a common decorative element in artwork from many cultures. It's interesting that something so commonplace can become the object of mathematical study, and that even though the questions that mathematicians ask about knots are often simple, such as "How can we tell if two knots are the same?", the answers can turn out to be surprisingly complex.

Connections to chemistry

Interest in comparing and tabulating knots actually grew out of a mistaken theory in chemistry. In the 1880s, chemists believed that space was filled with something called ether, and Lord Kelvin proposed that different atoms were different types of knots in the ether[1].

Belief in ether soon faded, but interest in identifying and cataloging different types of knots stayed. Knot theory soon grew into its own mathematical discipline, which for a while seemed to have nothing to do with chemistry. Recently, however, scientists have begun to see that large, complicated molecules are often knotted, and it looks as thought knot theory might once again be important to chemistry.

The Perko knots are sometimes used to illustrate the point that knot theory is still a relatively young mathematical field, and that it is accessible to non-mathematicians.


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References

  1. 1.0 1.1 1.2 1.3 Adams, C. (2004). The knot book: An elementary introduction to the mathematical theory of knots. Providence, RI: American Mathematical Society.
  2. Perko, K. (1974). On the classification of knots. Proceedings of the American Mathematical Society, 45, 262-266.
  3. Weisstein, Eric W. "Dowker Notation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DowkerNotation.html





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