The Unapologetic Mathematician

Mathematics for the interested outsider

Quandles

At long last I come to quandles. I know there are some readers who have been waiting for this, but I wanted to at least get through a bunch of group theory before I introduced them, because they tend to feel a bit weirder so it’s good to warm up before jumping into them.

The story of quandles really begins back in the late ’50s and early ’60s when John Conway and Gavin Wraith considered the wrack and ruin that’s left when you violently rip away the composition from a group and just leave behind its conjugation action. This is a set with an operation x\triangleright y=xyx^{-1}, and it’s already a quandle. The part of the structure they considered, though, has lost its ‘w’ and become known as a “rack”.

In 1982, David Joyce independently discovered these structures while a student under Peter Freyd working on knot theory with a very categorical flavor (hmm.. sounds familiar). He called them “quandles” because he wanted a word that didn’t mean anything else already, and when the term popped into his head he just liked the sound of it. There are other names for similar structures, but “quandle” is the one that really took hold, partially because there are a lot of unusual algebraic structures that aren’t good for much but their own interest, but “quandle” was the term the knot theorists picked up and ran with.

Actually, after hearing one of my talks Dr. Freyd mentioned that Joyce had come up with a lot of good things while a student, but he (Freyd) never thought much would come of quandles. In the end quandles have become the biggest thing to come out of his (Joyce’s) thesis.

Okay, so let’s get down to work. There are three axioms for the structure of a quandle, and I’ll go through them in the reverse of the usual order for reasons that will become apparent. We start with a set with two operations, written x\triangleright y and y\triangleleft x.

The third and most important axiom is that x\triangleright y distributes over itself: x\triangleright (y\triangleright z)=(x\triangleright y)\triangleright(x\triangleright z). A set with one operation satisfying this property is called a “shelf”, leading to Alissa Crans’ calling the property “shelf-distributivity”. No, I’m not going to let her live down making such an awful pun, mostly because she beat me to it. We can verify that conjugation in a group satisfies this property:
x\triangleright (y\triangleright z)=x(yzy^{-1})x^{-1}=(xyx^{-1})(xzx^{-1})(xy^{-1}x^{-1})=
=(xyx^{-1})(xzx^{-1})(xyx^{-1})^{-1}=(x\triangleright y)\triangleright(x\triangleright z)

The second axiom is that the two operations undo each other: (x\triangleright y)\triangleleft x=y=x\triangleright(y\triangleleft x). Some authors just focus on the one operation and insist that for every a and b the equation a\triangleright x=b have a unique solution. Our second operation is just what gives you back that solution. A shelf satisfying (either form of) this axiom is called a rack. We again verify this for conjugation, using conjugation by the inverse as our second operation:
(x\triangleright y)\triangleleft x=x^{-1}(xyx^{-1})x=y=x(x^{-1}yx)x^{-1}=x\triangleright(y\triangleleft x).

Finally, the first axiom is that x\triangleright x=x. Indeed, for a group we have x\triangleright x=xxx^{-1}=x. This axiom makes a rack into a quandle.

One more specialization comes in handy: we call a quandle “involutory” if x\triangleright y=y\triangleleft x. Equivalently (by the second axiom), x\triangleright(x\triangleright y)=y. That is, x acts on y by some sort of reflection, and acting twice restores the original.

As a bit of practice, check that in a rack the second operation is also self-distributive. That is, (z\triangleleft y)\triangleleft x=(z\triangleleft x)\triangleleft(y\triangleleft x). Also verify that if we start with an abelian group G (writing group composition as addition), the operation g\triangleright h=2g-h makes the set of elements of G into an involutory quandle.

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March 19, 2007 - Posted by | Quandles

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