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.
In 1942, Mituhisa Takasaki [ja] introduced an algebraic structure which he called a kei (圭),[1][2] which would later come to be known as an involutive quandle.
[3] His motivation was to find a nonassociative algebraic structure to capture the notion of a reflection in the context of finite geometry.
[2][3] The idea was rediscovered and generalized in an unpublished 1959 correspondence between John Conway and Gavin Wraith, who at the time were undergraduate students at the University of Cambridge.
Wraith had become interested in these structures (which he initially dubbed sequentials) while at school.
[4] 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.
These constructs surfaced again in the 1980s: in a 1982 paper by David Joyce[5] (where the term quandle, an arbitrary nonsense word, was coined),[6] in a 1982 paper by Sergei Matveev (under the name distributive groupoids)[7] and in a 1986 conference paper by Egbert Brieskorn (where they were called automorphic sets).
[8] A detailed overview of racks and their applications in knot theory may be found in the paper by Colin Rourke and Roger Fenn.
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.
The third and fourth rack axioms then say that these left and right actions are inverses of each other.
If we eliminate the right action and keep the left one, we obtain the terse definition given initially.
is by no means universal: many authors use exponential notation and while many others write 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: which are consequences of the definition(s) given earlier.
or equivalently Every group gives a quandle where the operations come from conjugation: 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 has a 'fundamental quandle'.
To define this, one can note that the fundamental group of the knot complement, or knot group, has a presentation (the Wirtinger presentation) in which the relations only involve conjugation.
The fundamental quandle is a very powerful invariant of knots.
Less powerful but more easily computable invariants of knots may be obtained by counting the homomorphisms from the knot quandle to a fixed quandle
Since the Wirtinger presentation has one generator for each strand in a knot diagram, these invariants can be computed by counting ways of labelling each strand by an element of
More sophisticated invariants of this sort can be constructed with the help of quandle cohomology.
made into a quandle with the left action given by Racks are a useful generalization of quandles in topology, since while quandles can represent knots on a round linear object (such as rope or a thread), racks can represent ribbons, which may be twisted as well as knotted.