Quandles

Since I have to begin somewhere, I might as well start with something I find interesting. Back in the summer of 2005, I participated in an REU; our research group (one of four or so) studied objects called quandles. I’ll go into the specifics of what we were trying to do in a later post — today I just want explain what a quandle is.

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 \vartriangleright : Q \times Q \rightarrow Q. We then impose the following axioms:

    \forall a,b,c \in Q the following holds:

  1. a \vartriangleright a = a;
  2. \exists x \in Q such that a \vartriangleright x = b; and lastly,
  3. (a \vartriangleright b) \vartriangleright c = (a \vartriangleright c) \vartriangleright (b \vartriangleright c).

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 \forall x \in \mathbb{Z}: x + x = x! This is certainly true for 0: 0+0 = 0. But for every other integer, it’s absured. The reason is that in things like groups we can prove:

Theorem:

Let (G, \cdot) be a group and let x \in G. Then x \cdot x = x \implies x = \text{id}_G.

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 Q, P then \forall p \in P: Q \cong Q \oplus \{p\}. 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. (a \cdot b) \cdot c = a \cdot (b \cdot c);

2. (a \vartriangleright b) \vartriangleright c = (a \vartriangleright c) \vartriangleright (b \vartriangleright c).

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.)

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4 Responses to Quandles

  1. Thanks for the reference. If you liked the run through group theory, you might like my own post about quandles

  2. Geoff says:

    Very nice post. I completely forgot you had covered quandles too.

  3. By the way, did you notice that the paper I posted the other day is mostly about quandles?

  4. Geoff says:

    I saw it when you first posted it, but I haven’t looked at your newest version. I’m hoping to read it if I have some time this week — it looks interesting, though I have a feeling the category-theoretic stuff will be over my head.

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