In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams.
While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group.
History
In 1942, introduced an algebraic structure which he called a (), which would later come to be known as an involutive quandle. His motivation was to find a nonassociative algebraic structure to capture the notion of a reflection in the context of finite geometry. Conway renamed them wracks, partly as a pun on his colleague's name, and partly because they arise as the remnants (or 'wrack and ruin') of a group when one discards the multiplicative structure and considers only the conjugation structure. The spelling 'rack' has now become prevalent.
These constructs surfaced again in the 1980s: in a 1982 paper by David Joyce (where the term quandle, an arbitrary nonsense word, was coined), in a 1982 paper by Sergei Matveev (under the name distributive groupoids) and in a 1986 conference paper by Egbert Brieskorn (where they were called automorphic sets). A detailed overview of racks and their applications in knot theory may be found in the paper by Colin Rourke and Roger Fenn.
Racks
A rack may be defined as a set <math>\mathrm{R}</math> with a binary operation <math>\triangleleft</math> such that for every <math>a, b, c \in \mathrm{R}</math> the self-distributive law holds:
<math display=block>a \triangleleft(b \triangleleft c) = (a \triangleleft b) \triangleleft(a \triangleleft c)</math>
and for every <math>a, b \in \mathrm{R}</math>, there exists a unique <math>c \in \mathrm{R}</math> such that
<math display=block>a \triangleleft c = b.</math>
This definition, while terse and commonly used, is suboptimal for certain purposes because it contains an existential quantifier which is not really necessary. To avoid this, we may write the unique <math>c \in \mathrm{R}</math> such that <math>a \triangleleft c = b</math> as <math>b \triangleright a</math>. We then have
<math display=block> a \triangleleft c = b \iff c = b \triangleright a, </math>
and thus
<math display=block> a \triangleleft(b \triangleright a) = b,</math>
and
<math display=block>(a \triangleleft b) \triangleright a = b.</math>
Using this idea, a rack may be equivalently defined as a set <math>\mathrm{R}</math> with two binary operations <math>\triangleleft </math> and <math>\triangleright</math> such that for all <math>a, b, c \in \mathrm R</math>:
- <math>a \triangleleft(b \triangleleft c) = (a \triangleleft b) \triangleleft(a \triangleleft c)</math> (left self-distributive law)
- <math>(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright(b \triangleright a)</math> (right self-distributive law)
- <math>(a \triangleleft b) \triangleright a = b</math>
- <math>a \triangleleft(b \triangleright a) = b</math>
It is convenient to say that the element <math>a \in \mathrm{R}</math> is acting from the left in the expression <math>a \triangleleft b</math>, and acting from the right in the expression <math>b \triangleright a</math>. The third and fourth rack axioms then say that these left and right actions are inverses of each other. Using this, we can eliminate either one of these actions from the definition of rack. If we eliminate the right action and keep the left one, we obtain the terse definition given initially.
Many different conventions are used in the literature on racks and quandles. For example, many authors prefer to work with just the right action. Furthermore, the use of the symbols <math>\triangleleft</math> and <math>\triangleright</math> is by no means universal: many authors use exponential notation
<math display=block>a \triangleleft b = {}^a b</math>
and
<math display=block>b \triangleright a = b^a,</math>
while many others write
<math display=block>b \triangleright a = b \star a. </math>
Yet another equivalent definition of a rack is that it is a set where each element acts on the left and right as automorphisms of the rack, with the left action being the inverse of the right one. In this definition, the fact that each element acts as automorphisms encodes the left and right self-distributivity laws, and also these laws:
<math display=block>\begin{align}
a \triangleleft(b \triangleright c) &= (a \triangleleft b) \triangleright(a\ \triangleleft c) \\
(c \triangleleft b) \triangleright a &= (c \triangleright a) \triangleleft(b \triangleright a)
\end{align}</math>
which are consequences of the definition(s) given earlier.
Quandles
A quandle is defined as an idempotent rack, <math>\mathrm{Q}</math>, such that for all <math>a \in \mathrm{Q}</math>
<math display=block>a \triangleleft a = a,</math>
or equivalently
<math display=block>a \triangleright a = a.</math>
Examples and applications
Every group gives a quandle where the operations come from conjugation:
<math display=block>\begin{align}
a \triangleleft b &= a b a^{-1} \\
b \triangleright a &= a^{-1} b a \\
&= a^{-1} \triangleleft b
\end{align}</math>
In fact, every equational law satisfied by conjugation in a group follows from the quandle axioms. So, one can think of a quandle as what is left of a group when we forget multiplication, the identity, and inverses, and only remember the operation of conjugation.
Every tame knot in three-dimensional Euclidean space (or more generally, a 3-sphere) has a 'fundamental quandle'