Knuth equivalence on a necklace

Lately, I’ve been working on some open problems related to a Young tableau operation called catabolism, which involves some interesting tableaux combinatorics. While working the other day, I encountered a simple and beautiful fact that I never would have expected.

Suppose we have a word $w$ whose letters are from the alphabet $\{1,\ldots,n\}$ (allowing repeated letters). A Knuth move is one of the following:

  1. Given three consecutive letters $xyz$ with $x>y$ and $x\le z$, replace $xyz$ with $xzy$, or vice versa (replace $xzy$ with $xyz$).
  2. Given three consecutive letters $xyz$ with $z\ge y$ and $z\lt x$, replace $xyz$ with $yxz$, or vice versa (replace $yxz$ with $xyz$).

In other words, if you have three consecutive letters and either the first or the third lies between the other two in size, the end letter acts as a pivot about which the other two letters switch places. Visually:

KnuthGood

In the case of a tie, think of the repeated letters as getting larger as you go to the right. For instance, in the word $211$, the rightmost $1$ is “larger” than the middle $1$, so $211$ can be replaced with $121$ by switching the left $1$ and the $2$.

If two words can be reached from each other by a sequence of Knuth moves, we say they are Knuth equivalent. For instance, $12143$ is equivalent to $24113$ via the sequence of moves:

12143
21143
21143
21413
21413
24113

In addition, not all words on the same letters are equivalent: the word $123456$, for instance, has no valid Knuth moves and so it is only equivalent to itself.

Since the moves are invertible, Knuth equivalence sorts all words into Knuth equivalence classes. In 1970, Donald Knuth realized that these equivalence classes are in one-to-one correspondence with semistandard Young tableaux, which has made them very useful in algebraic combinatorics.

Recently, I needed to understand what happens when you allow the Knuth moves to “wrap around at the edge”. In other words, think of the word as being on a necklace or counterclockwise-oriented circle, and allow the same Knuth moves as before. For instance, $123456$ is now equivalent to $623451$ since the $5$ is between the next two letters, $1$ and $6$.

Now, we get “cyclic” Knuth equivlance classes. How do these relate to the original Knuth equivalence classes? It turns out that there is only one cyclic Knuth equivalence class for any given collection of letters. That is, for any two arrangements of the same collection of letters around a necklace, there is a sequence of cyclic Knuth moves that takes one to the other.

It’s a nice fact – can you see why it’s true?

Circles of Apollonius… and magnetism!

These two concepts go together in a very natural way. Prepare for a breathtaking real-world application of Euclidean geometry!

Suppose you have two identical long wires side by side, parallel to each other and connected to each other at one end, and a current is flowing through one wire and back through the other in the other direction. As in this video, each wire generates a magnetic field, and the magnetic field forces the two wires towards each other. The question is, just before the wires start to move towards each other, what does the magnetic field look like?

Let’s simplify the problem a bit. Viewing the wires head-on, we can draw them as two points, and envision one current going into the page (red) and another current going out of the page (blue).

Some basic electrodynamics tells us that the magnetic field vectors generated by one of the wires alone are tangent to the circles centered at the wire, in the plane perpendicular to the wire. Furthermore, the magnitude of these vectors at a distance $r$ from the wire is proportional to $1/r$.

So, we can sketch the magnetic field arising from each wire separately, in two different colors, with the thickness and brightness indicating the strength of the field:

We can add these magnetic fields together to get a single magnetic field that looks like this:

The magnetic field lines appear to be circles! What are these circles?

A very similar-looking diagram recently came up in a class I was teaching on Inversive Geometry at the Math Olympiad Summer Program (MOP) this summer. A circle of Apollonius with respect to two given points $A$ and $B$ can be defined as any circle for which $A$ inverts to $B$ about that circle. Alternatively, and perhaps more simply, it is the locus of points $P$ having a given constant value of the ratio $PA/PB$.

One student in the class pointed out that these circles look very much like a magnetic field of some sort. Intrigued, I asked around and did a bit of research. I found that the circles of Apollonius arise as the equipotential lines of two different point charges.

However, this wasn’t a realization of the circles as a magnetic field. Still curious, I brought the question up at the lunch table the next day. One of my co-workers at MOP, Alex Zhai, suggested to try two parallel wires with opposite currents. Perhaps the magnetic field lines were the circles of Apollonius between the two points that correspond to the wires. It certainly looked correct. How could we verify this?