A rack is a set equipped with a binary operation satisfying axioms analogous to the second and third Reidemeister moves in knot theory. The most commonly appearing racks in nature are quandles, which satisfy an additional axiom analogous to the first Reidemeister move.
While mainly used to obtain invariants of framed knots, racks are interesting algebraic structures in their own right: they capture the idea of an algebraic structure where every element acts as an automorphism of that structure.
A rack may be defined as a set $R$ with a binary operation $\triangleright$ such that for every $a, b, c \in R$ the self-distributive law holds:
and for every $a,b \in R$ there exists a unique $c \in R$ such that
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 $c \in R$ such that $a \triangleright c = b$ as $b \triangleleft a$. We then have
and thus
and
Using this idea, a rack may be equivalently defined as a set $R$ with two binary operations $\triangleright$ and $\triangleleft$ such that for all $a, b, c \in R$:
It is convenient to say that the element $a \in R$ is acting from the left in the expression $a \triangleright b$, and acting from the right in the expression $b \triangleleft a$. 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. However, the longer definition makes it clear that racks are algebras of Lawvere theory.
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 $\triangleright$ and $\triangleleft$ is by no means universal: many authors use exponential notation
and
while many others write
We may equivalently define a rack in the following more conceptual manner: 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:
which are consequences of the definition(s) given earlier.
Even simpler than a rack is a shelf. A set with a binary operation $\triangleright$ obeying the left self-distributive law
is called a left shelf, and similarly a set with a binary operation $\triangleleft$ obeying the right self-distributive law is called a right shelf.
A unital left shelf is the same as a graphic monoid: for a proof see graphic category.
Gavin Wraith, A personal story about knots.
Wikipedia, Racks and quandles
Colin Rourke, Roger Fenn, (1992). Racks and links in codimension 2. J. Knot Theory and Its Ramifications 1 (4): 343–406.
Roger Fenn, Colin Rourke, Brian Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321–356 MR1364012doi; The rack space, Trans. Amer. Math. Soc. 359 (2007), no. 2, 701–740 MR2255194
Sam Nelson, Quandle theory.
J. Scott Carter, A survey of quandle ideas, arXiv.
Seiichi Kamada, Knot invariants derived from quandles and racks, arXiv.
Alissa Crans, Shelves, racks, spindles and quandles, arXiv, in Lie 2-Algebras, p. 56.
The last reference makes it clear that racks are algebras of a Lawvere theory, so that racks may be defined in any cartesian monoidal category (a category with finite products).
A generalization of Lie integration to conjectural Leibniz groups has been conjectured by J-L. Loday. A local version via local Lie racks has been proposed in
There are also crossed modules for racks: