Perko pair knots
|Perko pair knots|
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.
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.
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 movesAs was stated above, knots are considered to be the same if one can be rearra [...]
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 knot's 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:
Type II Reidemeister move:
Type III Reidemeister move:
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. 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 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.
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.
Belief in ether soon faded, but interest in identifying and cataloging different types of knots remained. Knot theory soon grew into its own mathematical discipline, which for many years seemed to have nothing to do with chemistry. Recently, however, scientists have begun to see that large, complicated molecules are often knotted, and have turned to knot theory to help understand them. Image 7 shows a knotted molecule. It now appears that knot theory is important to chemistry, although not in the way its founders thought.
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- Adams, C. (2004). The knot book: An elementary introduction to the mathematical theory of knots. Providence, RI: American Mathematical Society.
- Perko, K. (1974). On the classification of knots. Proceedings of the American Mathematical Society, 45, 262-266.
- Weisstein, Eric W. "Dowker Notation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DowkerNotation.html
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