While I still intend to (some day) continue with quandle posts, for the time being I’ll be switching to talking about the work I’m doing this summer (and probably will talk a bit about most of the work I did last summer, which is very tangentially related.)

You remember that hitting ^C (control-C) while mpg123 is playing a file will cause it to output something like

[56:36] Decoding of sterling_podcast.mp3 finished.

and then quit. What’s more, the -k option to mpg123 takes an integer n and starts playing the file it was passed n frames in. (For those of you unfamiliar with the structure of mp3 files, they store the audio data in segmented frames containing 26ms of audio data.) A quick invocation of Google tells you that 1 second of playback takes about 38.461 frames. So if you had a way to store mpg123′s output, you could use “mpg123 -k n /path/to/file” as a commandline mp3 player with rudimentary bookmarking.

But how do you store the output?

One of the great things about the commandline environment is its ability to pass output data around. One of the important elements of this flexibility is the “Standard Output,” which I’ll refer to stdout. For the uninitiated, stdout is where all the commandline programs send their output; usually, (and this is crucial) but not always, this is simply your terminal screen. However, it is possible to redirect this output to a file using (in Bash at least) the construct &>. An example:

The command “ls /home &> contents.txt” will write a list of the contents of the /home to the file contents.txt.

So there’s the answer. You can use

“mpg123 -k n /path/to/file &> .bookmark”

to play your long file, hit ^C to stop at any point, and be secure in the knowledge that you’ve got a bookmark waiting there for next time. If you wanted to be really fancy you could even write a simple script to pull the stopping point out of .bookmark and, after computing what n should be, launch mpg123.

Et voilà!

Suppose is a homomorphism. Then ; moreover, the map by is an isomorphism.

One of the points of the proof, since is defined on the cosets of by specifying the image of a representative, is whether or not the map is even well defined for some In particular, will taking different representatives of the coset always have the same image? It turns out that the map is indeed well defined – it’s a rather straightforward argument – but this just brings up another question:

If a map is well defined, does it have the property of well-definededness?

Yay, English!

EDIT: Fixed my accidental interspersing of for

This being my first mathy post, it’s probably obvious that quandles are a pet topic of mine. And no, I don’t like them just because they may have the best name in a technical field this side of quarks (future nomenclaturists take note – both begin with q.) Joking aside quandles fascinate me because, as we will learn in a moment, quandles are nonassociative. We’ll see, in later posts, that this makes quandles behave quite differently than objects like groups and vector spaces that we may be familiar with.

I’ll talk in more depth about the motivations for studying quandles at a later point; suffice it to say that quandles arise in a very natural way in the context of knot theory. For those of you familiar with knot theory, with the proper setup quandles encode the Reidermeister moves. Quandles also have a significant connection with the fundamental group of the knot complement, if I ever get to algebraic geometry I’ll talk about this connection.[1]

So what exactly are quandles? They are an algebraic system like groups. If you don’t know what a group is, I’ll get there eventually. In defining a quandle, we start with a set Q and a closed binary operation . We then impose the following axioms:

the following holds:
1. ;
2. such that ; and lastly,
3. .

For those of you familiar with more mainstream things like group theory, these axioms probably seem strange. We’ll start with axiom 1. This is like saying that This is certainly true for . But for every other integer, it’s absured. The reason is that in things like groups we can prove:

Theorem:

Let be a group and let . Then

I’m going to withhold the proof of this until the groups section, but if you’re familiar with groups, this is a very straightforward thing to prove.

Put group theoretically: in groups only the the identity element has order 1; in quandles every element has order 1. This axiom will give us some quaint strangeness. If you’ve no idea what the order of an element is – patience grasshopper (alternatively, hop over to John Armstrong’s excellent The Unapologetic Mathematician which, in addition to inspiring this blog, has a great write up of group theory from a while back.)

Axiom 1 causes some quaint strangeness like the fact that, give quandles then Axiom 3 causes strangeness that gives me headaches. It’s also what makes quandles so interesting – axiom 3 is when quandles become nonassociative. Compare the following:

1. ;

2. .

The first is familiar, it says something is associative – for example (3 + 4) + 5 = 3 + (4 + 5). The second is weird. It says that in quandles, our operation distributes over itself! That’s like saying (3 + 4) + 5 = (3 + 5) + (4 + 5). Ultimately, this is what drew me to quandles – the fact that such a weird structure would crop up very naturally in a number of places.

As always, if you spot any mistakes or would like references on any of the above, just ask. And sorry for the many forward references.

Up next: Examples of Quandles – The trivial quandle on N elements, the Tait-3 quandles, conjugation quandles, and more…

PS. As a side note, today I checked out Baby Rudin 2nd ed. from the science library today. Time permitting (which is a huge if this semester), and in spite of the fact I’m taking Advanced Calc right now, I’m planning on working my way through is – proving his theorems before reading his proofs, doing the problems, and just generally putting some mathematical hair in my beard. (For those of you that don’t know, Rudin’s Principles of Mathematical Analysis is held in quite high esteem by the many people who learned analysis from its dense but rich pages.) Expect updates. Funny, no?

[1] Some history: As far as I know, this connection (specifically, that there’s a presentation of the fundamental group of the knot complement with only conjugation relations – groups under conjugation form quandles as we’ll see later) was first established by David Joyce. While Joyce coined the name quandle and – to my knowledge – first established their topological usage, quandles were considered at least as early as the 1950′s by John Conway (who called them wracks and didn’t impose axiom 3.)

I study math because I legitimately enjoy the subject, hard as that may be to believe if you’re not familiar with upper level mathematics. I also study computer science because I like things like programming, operating systems, and databases. Yes, I run Linux. Beyond that… we’ll see where it goes.