May 21, 2015

The Origin of the Word “Quandle”

Posted by John Baez

A quandle is a set equipped with a binary operation with number of properties, the most important being that it distributes over itself:

$a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright c)$

They show up in knot theory, where the operation $\triangleright$ describes what happens when one strand crosses over another… and the laws are chosen so that the quandle gives an invariant of a knot, independent of how you draw it. Even better, the quandle is a complete invariant of knots: if two knots have isomorphic quandles, there’s a diffeomorphism of $\mathbb{R}^3$ mapping one knot to the other.

I’ve always wondered where the name ‘quandle’ came from. So I decided to ask their inventor, David Joyce—who also proved the theorem I just mentioned.

He replied:

I needed a usable word. “Distributive algebra” had too many syllables. Piffle was already taken. I tried trindle and quagle, but they didn’t seem right, so I went with quandle.

So there you go! Another mystery unraveled.

I wanted a short term, just one word, that wasn’t taken. I didn’t want to use a common word like group or ring which had a meaning in common English, so I selected a nonsense word. It just sounded good. I was thinking at the time that an alternative would be “conjugacy algebra,” but that was too long, and the word “algebra” suggests that addition and multiplication should be operations, which they aren’t in quandles.

Taken together, I hope they add up to the whole story.

Posted at May 21, 2015 7:41 AM UTC

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Re: The Origin of the Word “Quandle”

Cool! In a completely different setting there are operations which behave similarly (they distribute over themselves): weakening and substitution in type theory!

Taking an algebraic point of view of what type dependency is, one could model contexts as the objects of a category $C$, and families as the morphisms. And then there is a weakening structure which consists of functors $W_f : C/B \to C/A$ for any morphism $f:A\to B$, such that

$W_f/(h\circ g)\circ W_g = W_{W_A(g)}\circ W_f/h$

for any $f:A\to B$, $g:D\to C$ and $h:C\to B$.

This is just a way of saying that weakening distributes over itself, and I also think of it as saying that each $W_f$ preserves the weakening structure (on $C/B$). Forgetting about all the rest of the structure of type dependency, there is thus a notion of weakening system (which is a category with weakening structure), satisfying the same kind of distributivity as a quandle. And groups with conjugation form an example of such weakening systems.

Posted by: Egbert Rijke on May 21, 2015 9:55 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I don’t completely understand what you’re saying: for example, your equation involves expressions like $W_f/h$, where I don’t understand what the “$/h$” means.

But this is very interesting. If you explain it, we could perhaps create some topological invariant from a category with weakening structure. Even if not all the quandle axioms hold, there are things you can do with just self-distributivity.

Knotty logic!

Posted by: John Baez on May 21, 2015 10:06 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Hm, sorry. if $F$ is a functor from $C$ to $D$ and $X$ is an object of $C$, then I meant by $F/X$ the functor $F/X:C/X\to D/F(X)$ (which does what $F$ does on the arrows).

Because $W_f:C/B\to C/A$, and $C/B$ inherits the weakening structure from $C$, asking that $W_f$ preserves this weakening structure is asking that for any morphism $g : h\circ g \to h$ in $C/B$, $W_f$ commutes with $W_g$ in a sense (that it distributes over itself, but I had to take the slices into account).

Posted by: Egbert Rijke on May 21, 2015 10:27 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I’ll try to explain it better. In type theory there are judgments of the form $\Gamma\vdash A fam$ which say that $A$ is a family over $\Gamma$. That means that when you interpret type theory as an algebraic theory, you have a sort $ctx$ for contexts, and a sort $fam$ for families, and an operation $ft:fam\to ctx$ which says for each family what its context is. Now there’s also context extension $ext$, which is an operation $fam\to ctx$; and $ft$ and $ext$ behave like codomain and domain, respectively. The rest of the axioms of a category (identity morphisms and identity laws) we get from the type dependency rules too, so this forms a category.

This is a simplified perspective on B-systems, which Voevodsky has recently introduced, but he uses infinitely many sorts instead of extension. As in B-systems, there are operations for weakening and substitution, and there is a sort for terms as well. But let’s only focus on weakening.

When you weaken a family $B$ in context $\Gamma$ by a family $A$ in the same context $\Gamma$, you get a family $W_A(B)$ in context $ext(A)$, which is usually denoted by $\Gamma.A$. Moreover, the structure of type theory requires that $W_A$ is functorial in $B$. Since families are going to get interpreted as morphisms in this category, we get a functor $W_f: C/Y\to C/X$ for every $f:X\to Y$. This will actually be part of a functor from $C\to Cat$ such that an object $X$ gets mapped to $C/X$, but it’s not really neat to say that something is a functor into a two-category. This situation really asks for being treated as a pseudo-functor, but I preferred to stay close to the syntax of type theory.

So now we have these categories with weakening structures, and it follows that $C/X$ is also a category with weakening structure for each $X$ in a category $C$ with weakening structure. A homomorphism of such gadgets is a functor $F:C\to D$ such that for each morphism $f:X\to Y$ in $C$ we get

$F/X\circ W_f= W_{F(f)} \circ F/Y$

where $F/X:C/X\to D/F(X)$ is given by the arrow-part of $F$.

So it makes sense to require that each $W_f C/Y\to C/X$ is a homomorphism of categories with weakening structure, and this represents one of the properties of weakening in type theory.

More precisely, this requirement says that for each morphism $g:h\circ g\to h$ in $C/Y$, we have

$W_f/(h\circ g)\circ W_g = W_{W_f(g)}\circ W_f/h$.

Let’s just consider a simple case, where $h=1_{Y}$ and apply it to an object of $C/Y$, i.e. an arrow $k$ into $Y$. Then it says that

$W_f(W_g(k)) = W_{W_f(g)}(W_f(k)).$

So this requirement on weakening says exactly that weakening distributes over itself. So when I saw the law of quandles it reminded me of this :) I worked with Vladimir on this prespective on type theory, and he calls the full interpretation of type dependency in this way E-systems.

Posted by: Egbert Rijke on May 22, 2015 10:36 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

The history behind earlier names for quandles is recounted here: http://www.wra1th.plus.com/gcw/rants/math/Rack.html

Posted by: Joachim Kock on May 21, 2015 10:21 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Thanks Joachim! For many years I had pinned to my office door a portrait of myself drawn by you when you were very young. I was recognizable by the glasses, beard and blue book in my hand: DeMazure and Gabriel’s Groupes Algebriques. It is still on my bookshelf. Alas, the portrait became too dog-eared to survive the many retrenchments of retirement. Tak igen.

Posted by: Gavin Wraith on May 24, 2015 11:25 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

And yet another place where self-distributive binary operations come up is in large cardinal axioms! This comes about by studying the algebra of elementary embeddings between models of (fragments of) set theory, since many large cardinal axioms are formulated in terms of such elementary embeddings.

If $M$ is a model of (a suitably strong fragment of) set theory, then the elementary embeddings $M \to M$ admit one binary operation $\circ$ given by composition (which forms a monoid), but they also admit another operation $\cdot\,\,$, given* by application. This operation satisfies the left distributive rule

$j \cdot (k \cdot l) = (j \cdot k) \cdot (j \cdot l)$

simply because an elementary embedding between models of set theory must send application to application! Laver showed that if there is a nontrivial elementary embedding $j: V_\lambda \to V_\lambda$ (where $V_\lambda$ is the $\lambda$th von Neumann universe), then the algebra it generates under application is free subject to the left distributive law. And certain finite quotients of this free algebra are called Laver tables.

Apparently it was by thinking about these kind of things that Dehornoy first defined the standard ordering on the braid group, so this connection has even been quite fruitful!

*To be careful, we can’t apply $j$ to $k$ because $k$ is too big to be a set in $M$, so $j\cdot k$ needs to be defined by first restricting the domain of $k$ to be a set, then applying $j$, and then taking the union over these restrictions. This superficially relies on the idea that a function is a set of ordered pairs, but I think that reliance is only superficial.

Posted by: Tim Campion on May 22, 2015 12:51 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Thanks, Tim! This connection turns out to mean that large cardinal axioms are needed to prove some elementary-sounding results about quandles. Over on G+, David Roberts wrote:

Holy cow! There’s a theorem about the unboundedness of a reasonable-looking function $\mathbb{N} \to \mathbb{N}$ whose only known proof uses rank-into-rank embeddings. So what? you say. Well, these give the largest large cardinals we know that do not contradict ZFC! (as far as we know).

He cites this comment:

which says:

There’s an extremely elementary theorem whose only known proof relies on the existence of a rank-into-rank cardinal (basically the strongest large cardinal axiom not known to contradict ZFC).

Let $R_n = \mathbb{Z}/2^n\mathbb{Z}$.The $n$th Laver table is the unique binary operator $\star : R_n \times R_n \rightarrow R_n$ determined by the following conditions:

• $p \star 1 \equiv p + 1$
• $p \star (q \star r) \equiv (p \star q) \star (p \star r)$

Then the function $f_n(q) = 1 \star q$ is obviously periodic with some period $P(n)$ dividing $2^n$.

The existence of a rank-into-rank cardinal implies that $P(n)$ grows without bound.

Reference: Laver, Richard (1995), “On the algebra of elementary embeddings of a rank into itself”

Posted by: John Baez on June 3, 2015 1:19 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I am not a mathematician or physicist, nor do I even have one of those fancy university degrees, so this comment may be comparatively trivial, but the simplest occurrence of a self-distributive operation is perhaps a sort of analogue of union for (totally) ordered sets.

At least for finite things we have the following:

A list, or string, is a free monoid.

A bag, or multiset, is a free commutative monoid.

An ordered set is a free self-distributive monoid.

A set is a free commutative, self-distributive monoid.

ordered set : list :: set : multiset. etc.

In fact, we don’t need associativity, if we have a left/right identity and self-distributivity we already get associativity:

a·(b·c) = (a·b)·(a·c) = ((a·b)·a)·((a·b)·c) = ((a·b)·(a·1))·((a·b)·c) = (a·(b·1))·((a·b)·c) = (a·b)·((a·b)·c) = ((a·b)·1)·((a·b)·c) = (a·b)·(1·c) = (a·b)·c

So we can cancel any incidence of an argument in an expression if it already occurs in a further left position, a·b·a·c = a·b·c. So we see we are dealing with ordered sets.

I noticed these things some years ago when I wanted an analogue of regular expressions in which matched words were ordered by preference.

If we think of the elements of our algebra as operations, we can think of x·y as, “do x, if possible, otherwise do y” (and if y is also not possible the combined operation is considered impossible). The identity would then be an always impossible operation.

Posted by: Sam C on May 23, 2015 3:21 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

For some mysterious reason I’m just seeing your comment now, Sam. Your comment is fascinating in a couple of ways. First, I’d never seen this derivation of associativity for a self-distributive binary operation with a unit:

$\begin{array}{ccl} a \cdot (b \cdot c) &=& (a \cdot b) \cdot (a \cdot c) \\ &=& ((a \cdot b) \cdot a) \cdot ((a \cdot b) \cdot c) \\ &=& ((a \cdot b) \cdot (a \cdot 1)) \cdot ((a \cdot b) \cdot c) \\ &=& (a \cdot (b \cdot 1)) \cdot ((a \cdot b) \cdot c) \\ &=& (a \cdot b) \cdot ((a \cdot b) \cdot c) \\ &=& ((a \cdot b) \cdot 1) \cdot ((a \cdot b) \cdot c)\\ &=& (a \cdot b) \cdot (1 \cdot c) \\ &=& (a \cdot b) \cdot c \end{array}$

I’ll have to stare at this a few days to make sure it’s not a trick. How in the world did you think of this?

Second, while I don’t yet understand the relation between ordered sets and free self-distributive monoids, I can’t help but hope that this is related to what Tim Campion said in this discussion about cardinals and self-distributive binary operations… since cardinals are (certain especially nice) well-ordered sets, albeit typically infinite.

Posted by: John Baez on June 3, 2015 2:11 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Actually, the identity can be both arrived at or discovered and proven by an automated process that I will describe here. There’s no need for creativity or cleverness.

Let S be the set of statements of interest. Here, we’ll take S = {A,B,C,D} with the identities A: x = xe; B: x = ex; C: (xy)z = x(yz) and D: x(yz) = (xy)(xz).

Let M be any set of interpretations (or models) for the operations and T: S x M -> 2 the “semantics” function. The homomorphism from the free lattice generated by S to 2^M (which T(s,_) maps each s in S into) is a prime filter generated by a statement which, when expressed in conjunctive normal form, yields a set of CONJECTURES.

Now apply your favorite theorem prover to each conjecture. Whatever cannot be proven: find counter-examples, add them to M and re-index. This can be done incrementally. So you can actually just start out with M = {}, or M = a large standard set of automatically generated models.

For what follows let a,b,c,d denote the negations respectively of A,B,C,D; denote conjunctions by AB = A and B, disjunctions as A|B = A or B.

I won’t describe here the incremental process but the one that’s all-at-once – namely: run down all the tables of sizes 1,2,3,4 on up. Every combination except ABcD and aBCd shows up which immediately yields two conjectures: ABD -> C and BC -> A|D. A theorem prover on ABD -> C yields the desired result.

A random model search done in conjunction with a theorem prover for BC -> A|D may have difficulty resolving the issue … thus revealing the unexpected importance of those structures that satisfy BC alone. The conjecture is saying that BC structures tend to want to be either quandles or monoids. Betcha didn’t see that one coming.

There is a table of size 5 that resolves the conjecture. I will let you try and find it. Here’s a hint: look up syntactic nonoids and try to characterize the aBD structures.

The “semantic lattice” L(S) associated with S = {A, B, C, D} is the free lattice 2^2^S modulo the relation (ABD -> C) = true. This is the one and only non-trivial relation between the identities.

Conjectures, counter-examples are created automatically, theorems and proofs and established automatically. This is a Boolean Sieve.

Along the way, a second structure is “discovered”, namely, the one just discussed here.

In essence, what I just described here is a general framework that automates what mathematicians do. Slap on a natural language generator and it can spruce it up with the narratives, and it can write the book for you too.

Posted by: Rock Brentwood on September 7, 2016 1:45 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

That should read: “try to characterize the aBC structures”. If you want a more ambitious exercise – one which I’ve already quietly begun to carry out – take ALL the identities of interest in ALL the theories you can think of and throw them all together in the pot. Even better: generate every possible identity up to a given size; using various criteria to “normalize” the set … for instance: the identity a = bb can be replaced by a = aa and aa = bb. Run the process on them all and try to derive the entire taxonomy of modern algebra from it all.

In a comment further below I alluded to the fact that the axiomatization for the ternary algebra I used with affine geometry (which contains a quandle structure in it under certain conditions) was derived in part by an automated process. This was the process used.

If you do this with arithmetic; you can also find the identities to throw in the pot by starting out with the exhaustive list and systematically winnowing it down. Upon application of this process you will recover the entire stratification of modern algebra (i.e. groups, rings, semirings, fields, division rings, etc.)

Posted by: Rock Brentwood on September 7, 2016 1:53 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

If I may break it down a bit:

$a \cdot b = a \cdot (1 \cdot b) = (a \cdot 1) \cdot (a \cdot b) = a \cdot (a \cdot b)$

and similarly

$a \cdot b = (a \cdot b) \cdot a$

So now

\begin{aligned} a \cdot (b \cdot c) &= (a \cdot b) \cdot (a \cdot c) \\ &= ((a \cdot b) \cdot a) \cdot ((a \cdot b) \cdot c) \\ &= (a \cdot b) \cdot ((a \cdot b) \cdot c) \\ &= (a \cdot b) \cdot c \end{aligned}

As for coming up with this …

Posted by: andrew hubery on June 3, 2015 2:21 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Andrew wrote:

If I may break it down a bit:

$a \cdot b = a \cdot (1 \cdot b) = (a \cdot 1) \cdot (a \cdot b) = a \cdot (a \cdot b)$

and similarly

$a \cdot b = (a \cdot b) \cdot a$

Great!

I’m wondering what sort of algebraic structure we really have here. Alissa Crans has called a set with a left-distributive binary operation is a left shelf. We’re starting with a unital left shelf: one that has an element $1$ with $1 \cdot a = a = a \cdot 1$.

But then Sam C showed that such a thing is also a monoid: the binary operation is associative! And you’ve shown that it also obeys

$a b = a a b$

(though of course you did this as part of simplifying the proof that the binary operation is associative).

Taking $b = 1$, we see

$a^2 = a$

so it’s also a monoid where every element is idempotent.

Now, I’d really be happy if it were a commutative monoid where every element is idempotent, because this would demystify the law

$a b c = a b a c$

$a b c = a^2 b c = a b a c$

In short: any commutative monoid where every element is idempotent is a unital left shelf. But is the converse true?

Can someone deduce the commutative law from the unital left shelf laws? Or can someone find a counterexample: a noncommutative unital left shelf?

To conclude, I still don’t understand what Sam C said about the relation between unital left shelves and ordered sets. (I’ve been asleep for the last 8 hours.)

But a very nice sort of partially ordered set is a inf-semilattice: one where every finite set has a greatest lower bound. Such a thing is secretly the same as a commutative monoid where every element is idempotent… and so, it gives a unital left shelf.

How does this work? Well, suppose we have an inf-semilattice. The greatest lower bound of $a$ and $b$ is an element $a \vee b$. The greatest lower bound of the empty set is a ‘top’ element $\top$, greater than equal to all the rest. And the operation $\vee$ is commutative and associative, with $\top$ as unit. So we get a commutative monoid, and clearly every element is idempotent.

Conversely, given a commutative monoid where every element is idempotent, we can define $a \le b$ to mean $ab = a$, and check that this defines an inf-semilattice.

Posted by: John Baez on June 3, 2015 4:03 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Thanks for the replies John. I am glad that you found my comment interesting.

“I still don’t understand what Sam C said about the relation between unital left shelves and ordered sets.”

I did not state what I meant very well. Here is the nLab page on lists and free monoids: http://ncatlab.org/nlab/show/free+monoid

Adapting this to the case of unital left shelves:

Given a set S, the free unital left shelf on S is the set of all subsets of elements of S with a total ordering on their elements.

So we might define ordered sets of Xs as an inductive type by saying: An ordered set of Xs is either the empty ordered set {}, or a singleton ordered set {x} where x ∈ X, or the “ordered union” of two ordered sets X·Y.

And which must obey the unital left shelf laws:

{}·X = X·{} = X

X·(Y·Z) = (X·Y)·(X·Z)

To see that we are dealing with ordered sets notice that any expression involving the elements of a set S, a unit, and a binary operator obeying the unital left shelf laws has a normal form where each element in the expression appears only once (and the unit doesn’t appear). This is because, as I said before, we can cancel any element which already appears further left.

(a·b…·c·d)·x·(e·f…·g·h)·x·(i·j…·k·l) = (a·b…·c·d)·x·(e·f…·g·h)·(i·j…·k·l)

and, of course, our (a·b…·c·d),(e·f…·g·h), and (i·j…·k·l) may be empty since we can put the identity in their place if they are.

So, for example, the expressions on a free unital left shelf over the Booleans are equal to either 1, T, F, T·F, or F·T. That is, isomorphic to the totally ordered subsets of the Booleans.

Sorry about the formatting of my equations, I didn’t figure out how break them up onto separate lines.

Posted by: Sam C on June 3, 2015 6:33 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I forgot to mention regarding the normalisation, that the self-distributivity doesn’t reorder any elements with respect to their left-most occurrences; if the left most x is to the left of the left most y in a·b·c then it is in a·b·a·c and vice-versa.

Posted by: Sam C on June 3, 2015 9:45 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Apparently there are noncommutative unital left shelves. Last night I emailed James Dolan, and I just read the reply:

JB: Did you know this? Say you have a binary operation that distributes over itself on the left and has an element 1 that’s a left and right unit. Then it’s associative!

JD: not completely sure whether i knew it ….

JB: I don’t know what this means, except of course that having such a unit is a weird thing.

JD: might it be that it means that you’ve got a semilattice, aka acommutative monoid with all elements idempotent? self-distributivity (xy)z=(xz)(yz) follows there from idempotence and middle-four exchange: (xy)z = (xy)(zz)=(xz)(yz).

so can we get back from “self-distributive with 2-sided unit element” to “semilattice”?

hmm, i guess not. z = 1z = (11)z = (1z)(1z) = zz so indeed any element is idempotent, but there seem to be noncommutative examples. so the fact that we don’t seem to be getting just semilattices is making me suspect that i didn’t ever completely work this out before ….

at first i thought that there were examples involving otto’s law xyz=xz, but that seems to be on the wrong track ….

on the other hand, maybe the variety of monoids that you’re getting is precisely lawvere’s “graphic monoids” (satisfying xyx=yx) from that hegelian taco stuff …. which i should probably try to understand sometime, not having understood too much about it back when he talked about it at buffalo …. i vaguely suspect that it’s one of those sorts of things that i didn’t have the prerequisite mind-frame for back then, but might make perfect sense now …. hmm, particularly if there really is some nontrivial relationship to coxeter groups as some of the google links seem to be hinting ….

hmm, i wonder if the murphy monoid is a graphic monoid …..

The Murphy monoid is discussed here:

and the Hegelian taco is discussed here:

• F. William Lawvere, Display of graphics and their applications, as exemplified by 2-categories and the Hegelian “taco”, Proceedings of the First International Conference on Algebraic Methodology and Software Technology, University of Iowa, May 22-24 1989, Iowa City pp. 51–74.

Someone should put this latter paper online—I’ve never read it! I think it contains a 2-categorical diagram shaped like a taco; on the $n$Lab it’s summarized thus:

contains the most explicit expression of Hegelian concepts in the context of the presheaf topos on $\Delta_1$ and a footnote on Lawvere’s relation to Hegelian idealism.

Graphic monoids are discussed here:

Posted by: John Baez on June 3, 2015 4:21 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I see you’ve already figured it out, but for next time, Prover9/Mace4 is an excellent pair of tools to study these first-order questions. With an input file containing:

formulas(theory).

% Self-Distributivity
all x all y all z (x * (y * z) = (x * y) * (x * z)).
% There is a left and right identity element
exists e ((all x (e * x = x)) & (all y (y * e = y))).
% Non-commutativity
exists a exists b (a * b != b * a).

end_of_list.


Mace4 spits out the 3-element model you gave in 0.01 seconds.

Posted by: Ulrik Buchholtz on June 3, 2015 8:03 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Cool! I’m not sure I’d want the answer that fast — sitting around and thinking can be very helpful and also fun. But if I got stuck on a harder question of this type, an oracle of this sort could be useful.

Posted by: John Baez on June 3, 2015 8:32 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

A graphic monoid is one obeying the identity

$x \cdot y \cdot x = x \cdot y$

We have

Theorem. A unital left shelf is the same as a graphic monoid.

Proof. On the one hand, Sam C and Andrew Hubery showed above that any unital left shelf obeys the associative law and the graphic identity holds. The graphic identity works like this:

$x \cdot y = x \cdot (y \cdot 1) = (x \cdot y) \cdot (x \cdot 1) = x \cdot y \cdot x$

On the other hand, in any graphic monoid, the left self-distributive law holds:

$x \cdot (y \cdot z) = (x \cdot y) \cdot z = (x \cdot y \cdot x) \cdot z = (x \cdot y) \cdot (x \cdot z) \qquad \qed$

A commutative graphic monoid is the same as a semilattice. (In a previous comment I was taking this to be an inf-semilattice, but one can simply say ‘semilattice’.)

However, there are also plenty of noncommutative examples. The simplest one has 3 elements, $\{1, \delta_1, \delta_2\}$ with

$\delta_i \delta_j = \delta_i$

and the category of presheaves on this monoid is equivalent to the category of reflexive graphs, as mentioned on the nLab.

Posted by: John Baez on June 3, 2015 5:51 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Let me just say what the 3-element monoid

$\qquad M =\{1, s, t \}$

with

$s t = s s = s$

$t s = t t = t$

has to do with reflexive graphs. My change in notation is supposed to make it more obvious.

Suppose you have a category and you don’t want to talk about its objects, only its morphisms. Then for any morphism $f : x \to y$ we could define its source $s f$ to be the identity morphism $1_x$, instead of the object $x$. Similarly, we could define its target $t f$ to be the morphisms $1_y$.

With these definitions we have two operations that carry morphisms to morphisms, $s$ and $t$, and they obey the rules

$t s = s s = s$

$s t = t t = t$

You’ll notice the multiplication rules are backwards from what we had above. So, if you think of a monoid as a 1-object category, we’re getting a functor

$F: M^{op} \to Set$

sending the one object of $M^{op}$ to the set of morphisms in our category, and $s,t$ to the operations defined above.

This is just a presheaf on $M$.

But in fact none of this requires having a category: it works whenever we have a reflexive graph: that is, a directed multigraph where each node $x$ has a specified edge $1_x : x \to x$$. So, any reflexive graph gives a presheaf on$M$. And conversely, any presheaf on$M$defines a reflexive graph. (We usually think of reflexive graphs as presheaves on a category with two objects,$V$for vertices and$E$for edges. But this category is the Cauchy completion of the monoid$M\$, so their presheaf categories are equivalent.)

Posted by: John Baez on June 3, 2015 8:47 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Just a quick remark: I wouldn’t agree with the following.

quandles capture the essence of how the strands of a knot cross over each oher.

I would say that they capture algebraically one way to construct invariants from a knot diagram. In particular, I would say the following is the whole point of the definition, i.e., the definition of a quandle is set up for the following to be true.

yet they manage to give an invariant of a knot, independent of the way you draw it.

As with any diagrammatic invariant in knot theory, one needs Reidemeister’s theorem that isotopy of knots is captured by the Reidemeister moves on knot diagrams. But this has nothing specifically to do with quandles. The notion of a quandle is then exactly intended to ensure that the ways in which the arcs of a knot diagram can be labelled (by numbers modulo an integer, or more sophisticated things), is unchanged under Reidemeister moves on that knot diagram, i.e., such that we have an invariant.

Posted by: Richard Williamson on May 23, 2015 7:54 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I think your phrase “exactly intended” is saying what I meant when I said “capture the essence”. Namely, the quandle axioms are just a distillation of the Reidemeister moves.

Posted by: John Baez on June 3, 2015 9:02 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

But my point is precisely that the quandle axioms are not a distillation of the Reidemeister moves per se. They are a distillation of how labels of the arcs of a knot diagram must be chosen in order to ensure that the possible labellings of a knot diagram are invariant under the Reidemeister moves. These are two very different things: invariance under the Reidemeister moves can, of course, be regarded as the essence of knot theory, but labelling of arcs is only one of many ways to detect this invariance.

I was also responding to the fact that it might seem from what you wrote that it is a surprise that quandles give rise to knot invariants, whereas this is immediate from, and the very point of (with regard to knot theory, at least), the definition.

Posted by: Richard Williamson on June 4, 2015 11:33 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Richard wrote:

But my point is precisely that the quandle axioms are not a distillation of the Reidemeister moves per se. They are a distillation of how labels of the arcs of a knot diagram must be chosen in order to ensure that the possible labellings of a knot diagram are invariant under the Reidemeister moves.

Okay, that’s right. I wasn’t trying to be very precise; this is more precise.

Posted by: John Baez on June 4, 2015 4:04 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Two remarks on quandles/racks:

1. There are two simple maps of algebraic theories Rack -> Group. One (call it Conj) takes the rack operation x^y to the group operation of conjugation, y^(-1)xy, and this seems to be the most popular. However, the one that originally got me going (call it Core) maps the rack operation to the group operation xy^(-1)x, which does not seem to have drawn so much attention. The picture is “travel from x to y and keep going the same distance”. In other words, a rack is a sort of symmetric space. Given two elements a, b the set of elements x such that a^x = b forms a subrack, that we might call the equator of a and b.

2. Lassos versus curtainrings. Given a link in a 3-manifold, and a basepoint disjoint from it, the fundamental group of the complement of the link consists of homotopy classes of lassos, i.e paths beginning and ending at the basepoint that are tangled with the link. The elements of the fundamental rack, however, consist of homotopy classes of paths beginning at the basepoint and ending at a “curtainring” that slides on the link. There is an evident map of racks from the fundamental rack to the fundamental group (with conjugation) got by exploding the curtainring to a lasso. That this map is not necessarily injective can be seen by considering a sum of two knots, with the curtainring placed on the section connecting the two summands. As a curtainring it can only move elsewhere by going through one of the summands, getting its tail entangled in the process. A lasso, on the other hand, can be expanded to move over the outside of the summands, without the tail getting entangled.

Sorry, it needs a picture, with colours, and I am not sufficiently tech-savvy for that.

Posted by: Gavin Wraith on May 24, 2015 1:12 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I don’t know if when you say, “does not seem to have drawn so much attention” you mean so little attention that any example may be of interest, and I have no way of knowing the likelihood of your already knowing about, or being interested in, what I am about to say, but after I read some work of Vaughan Pratt’s pertaining to programming, I poked around looking at some more recent stuff he has done, all beyond my understanding, and I do recall he discusses just such an operation as you are talking about “travel from x to y and keep going the same distance”.

Posted by: Sam C on June 4, 2015 8:27 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Gavin Wraith, maybe I have misunderstood your interest, I am not saying Pratt’s work pertains to a mapping Rack -> Group, merely that it pertains to a “travel from x to y and keep going the same distance” operation.

Posted by: Sam C on June 4, 2015 8:35 AM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

As Gavin hinted, a “symmetric space” is a set equipped with a binary operation that lets you “go from $y$ to $x$ and keep going the same distance”, reaching a point $x \triangleright y$.

I usually call this operation “reflecting $y$ across $x$”. People often call the map

$y \mapsto x \triangleright y$

the “inversion symmetry about $x$”.

A good example of a symmetric space is a sphere: there’s an obvious inversion symmetry about any point $x$. It fixes the point $x$, and reflects any other point across $x$.

As Gavin noted, we can take any group and define

$x \triangleright y = x y^{-1} x$

If you ponder the formula, you’ll this is a way to reflect $y$ across $x$. Moreover, this operation obeys all the axioms of an involutory quandle:

• left self-distributivity: $x \triangleright (y \triangleright z) = (x \triangleright y)\triangleright (x \triangleright z)$

• idempotence: $x \triangleright x = x$

• involutoriness: $x \triangleright (x \triangleright y) = y$

In my opinion, the concept of involutory quandle is the most general, most beautiful concept of symmetric space! Usually people consider a more restrictive definition where the symmetric space is a manifold. In this definition we start with a Lie group $G$ and a subgroup $H$ that consists of the fixed points of some involution $\sigma : G \to G$: that is, a Lie group homomorphism with $\sigma^2 = 1$. Then $G/H$ becomes an involutory quandle, where we use the involution to define the operation $\triangleright$.

But clearly (to my mind) this should be a theorem about how to construct symmetric spaces, not the definition. And indeed, one can show that an involutory quandle in the category of smooth manifolds arises via this construction… if an extra condition holds: no point sufficiently near $x$ is a fixed point of reflection about $x$.

(For an example of an involutory quandle that doesn’t obey this extra condition, take any manifold and define $x \triangleright y = y$. Then every point is a fixed point of reflection about $x$.)

All this is great if you like differential geometry: symmetric spaces of this sort are wonderful things. But we can easily generalize the usual group-theoretic definition of symmetric space by letting $G$ and $H$ be arbitrary groups and letting $\sigma$ be a group homomorphism with $\sigma^2 = 1$. In this situation $G/H$ always gets the structure of an involutory quandle.

There should be some theorem saying that all involutory quandles obeying some extra conditions arise this way, but I don’t know it.

For details, see:

• Wolgang Bertram, The Geometry of Jordan and Lie Structures, Lecture Notes in Mathematics 1754, Springer, Berlin, 2000.

The relation to quandles is given in Theorem I.4.3.

Posted by: John Baez on June 4, 2015 4:53 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

I guess this means that the “reflecting y across x” operation itself has not not drawn much attention. But just in case anyone is interested, Pratt’s work that I was thinking of was:

Geodesic spaces: Euclid’s five postulates as an equational theory, starting with the second

Euclidean and non-Euclidean algebra

geodesic spaces : momentum :: groups : symmetry

Those are all links to PDFs of slides (there don’t seem to be any papers). They can all be found at Pratt’s homepage

John, I assume my earlier reply to you cleared up what I meant by the connection between unital left shelves and orderd sets?

Lastly, thanks to whoever fixed up that earlier reply and deleted my extraneous comment.

Posted by: Sam C on June 4, 2015 7:47 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Sam C wrote:

I guess this means that the “reflecting y across x” operation itself has not not drawn much attention.

It’s drawn a huge amount of attention in the the theory of symmetric spaces, which is all about this operation.

I believe Gavin meant is that people studying quandles (e.g. getting knot invariants from quandles and seeing what you can do with those invariants) have focused more attention on quandles that arise from groups via conjugation:

$x \triangleright y = x y x^{-1}$

than those that arise from groups via relection:

$x \triangleright y = x y^{-1} x$

or those that come from other symmetric space.

That seems roughly right. The Alexander polynomial, a very famous knot invariant, comes (a bit indirectly) from a group via reflection. However, not many people seem to talk about this fact.

Posted by: John Baez on June 4, 2015 8:40 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Sam C wrote:

John, I assume my earlier reply to you cleared up what I meant by the connection between unital left shelves and ordered sets?

Yes, I get it now—that’s very nice! Since unital left shelves are the same as monoids obeying the graphic identity $x y x = x y$, I added a remark about this to the $n$Lab page on graphic categories. I didn’t cite you because… well… I felt funny citing “Sam C” without a specific last name. But you can go in there and edit the page to add a citation if you want. Here’s what I said:

The free graphic monoid on a set $X$ is the set of totally ordered finite subsets of $S$, where the product of finite subsets $S$ and $T$ is the union $S \cup T$ ordered in such a way that the inclusions $S \hookrightarrow X$, $T \hookrightarrow X$ are order-preserving and all the elements of $T$ not in $S$ are greater than all elements of $S$. This is easy to understand from an example. Consider the free graphic monoid on $X = \{a,b\}$. This contains:

1 (corresponding to the empty subset of $X$),

$a$ (corresponding to the subset $\{a\}$),

$b$ (corresponding to $\{b\}$),

$a b$ (corresponding to $\{a,b\}$ ordered so that $a \lt b$),

$b a$ (corresponding to $\{a,b\}$ ordered so that $b \lt a$),

and nothing else, since for example $a b b = a b$ and $b a b = b a$ (since the union of ordered subsets $\{a,b\} \cup \{b\} = \{a, b\}$ inherits the same ordering that $\{a,b\}$ had).

Posted by: John Baez on June 4, 2015 8:47 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

John wrote,

The free graphic monoid on a set X is the set of totally ordered finite subsets of S, where the product of finite subsets S and T is the union S∪T ordered in such a way that the inclusions S↪X, T↪X are order-preserving and all the elements of T not in S are greater than all elements of S.

If I am understanding correctly, the inclusion T↪S∪T (you wrote T↪X, but I think this is what you meant) is not necessarily order-preserving.

S = {b}

T = {a,b}, a<b

S∪T = {b,a}, b<a

since b·(a·b) = b·a

So, I think

The free graphic monoid on a set X is the set of totally ordered finite subsets of X, where the product of finite subsets S and T is the union S∪T ordered in such a way that the inclusion S↪S∪T is order-preserving, and the inclusion T↪S∪T is order-preserving for all elements of T not in S, and all the elements of T not in S are greater than all elements of S.

Posted by: Sam C on June 4, 2015 11:33 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Thanks, you’re right. I’ll fix the nLab entry.

Posted by: John Baez on June 5, 2015 4:47 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

Let us use $\vec\amalg$ to denote the ordered disjoint union of ordered sets, putting the elements of the second term after those of the first.

Then Sam C’s union operation is

$S\cdot T=S\vec\amalg\left(T\setminus S\right)$, which looks as though it has a family resemblance to conjugation.

In fact, I think $\setminus$ here is a right inverse but not a left inverse to $\vec\amalg$?

Posted by: Tim Silverman on June 5, 2015 5:51 PM | Permalink | Reply to this

Re: The Origin of the Word “Quandle”

The commutative variety of a quandle has the same properties as an affine geometry over a characteristic 3 field; where the product operation is just the midpoint operator (a + b)/2 = (a + b)/-1 = -a + -b.

Posted by: Rock Brentwood on September 2, 2016 12:35 AM | Permalink | Reply to this

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