So, you're wasting time with the internet...

(If you're here, you must be!)

... and this Batty fellow keeps intruding with things you tried to forget that you never learned because of how terribly dull your course on matrices in linear algebra was, or is...

So, yeah, I do topology. In fact, algebraic topology. One of the things I like about algebraic topology is that sometimes you get to draw nice pictures. OK, so I can't draw worth your trouble, but it's fun anyways. Maybe Emacs or someone will make better pictures.

One more nifty thing about the pictures you can draw in topology is that they're useful for other parts of math, too, like group theory. And this is a big deal! There are some Nice and Easy-to-Describe topological spaces that "know" how to calculate things that we can't calculate. This is part of what makes algebraic topology difficult, but also part of why it's exciting. And if we're lucky, sometimes they give us help understanding the things we can calculate. In particular, being able to describe a calculation by drawing a picture can help us lots.

That's my introduction to

For instance, if you ever had to do linear algebra with matrices, then you probably heard about their determinants; for instance, a two-by-two matrix $\left(\begin{array}{cc}A&B\\C&D\end{array}\right)$ has determinant $AD - BC$, and a three-by-three matrix has an even worse determinant expression with perhaps six terms in it, and a four-by-four... they would have given you, however, the clever abbreviation

$$ \left| \begin{array}{cccc} a_{00} & a_{01} & a_{02} & a_03 \\a_{10} & a_{11} & a_{12} & a_{13} \\a_{20} & a_{21} & a_{22} & a_{23} \\a_{30} & a_{31} & a_{32} & a_{33}\end{array}\right| = a_{00} \left|\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right| - a_{01}\left|\begin{array}{ccc} a_{10} & a_{12} & a_{13} \\ a_{20} & a_{22} & a_{23} \\ a_{30} & a_{32} & a_{33}\end{array}\right| + {} $$
$$ a_{02} \left|\begin{array}{ccc} a_{10} & a_{11} & a_{13} \\ a_{20} & a_{21} & a_{23} \\ a_{30} & a_{31} & a_{33}\end{array}\right| - a_{03} \left|\begin{array}{ccc} a_{10} & a_{11} & a_{12} \\ a_{20} & a_{21} & a_{22} \\ a_{30} & a_{31} & a_{32}\end{array}\right| $$

and more things besides; but if you were lucky they also mentioned that there was a group acting on the set $\{0,1,2,3\}$, and that this group had a representation $\sigma$ as multiplication by $\pm 1$... if you were even luckier, they would have told you why this is true, and so forth --- especially, that this representation $\sigma$ tells you which sign to use for which terms when you expand the determinant all the way down.

Here are some pictures relating to these absurd claims.

This is an abstract of expanding the determinant of a (permutation) matrix. It looks like a forrest of lollipops! First you draw the circles corresponding to the permutation and drop the stem to plant the lollipop. Then you fill-in the remaining spaces (including the circles) in each row from left to right alternating between $+$ and $-$, and you always start with $+$! Because that's what the determinant expression says to do. Then you count how many $-$s you circled, and if you get an odd number, your permutation is odd; if you get an even number, the permutation is even. Easy, yes?

Here's another way:

Here you just write out 1,2,3,4,5,6,7 and draw an arc from each number in the first row to its image in the second row. Be careful that each arc is always going slightly down, so it doesn't cross itself, and each time you draw a new arc, make sure it doesn't hit any crossings you've already drawn! (If you want to break this rule, you have to be very careful in a more complicated way, so I won't allow it). Then you count how many crossings there are. If there's an odd number of crossings, then your permutation is odd. If there's an even number ... this should sound familiar by now. The point is, though, that these two procedures give the same notion of odd/even! Although, one is much easier to work with.

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<take a breath>

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See, the point of group operations is that you can compose them. If you've got permutations $g$ and $h$, there are more permutations related to them: $gh$ and $hg$ and ...
And both the matrices and the string pictures also represent composition; to compose permutations given their matrices, just multiply the matrices.

To compose permutations given their string pictures, just stack one string picture over the other!

And, finally, odd/evenness also represents (something of) the permutations: if $g$ and $h$ are even, then $gh$ is also even; the same if they're both odd: $gh$ is even again. If $g$ is even and $h$ is odd, then $gh$ is odd, and so is $hg$. It's easier to remember if you write it this way, though:

$\times$	$1$	$-1$
$1$	$1$	$-1$
$-1$	$-1$	$1$

Anyways, trying to look at the lollipops... what on earth are we to do!? The procedure for deciding between even and odd, it may be easy, but it obscures something of what's going on. In fact, I'd rather call it an obfuscation.

But in terms of strings, it's easy to check that stacking the pictures, as far as counting crossings is concerned, coresponds to addition.

It's that easy!
Even better, the string pictures suggest why we only care about even/oddness, and not about the full number of crossings.

Incidentally, while getting this sketchy little post into some sort of shape, I have figured out what I think is the right way to view the lollipop diagrams. You see, there is something it really is obfuscating. If you see it... !!!