In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, racks and quandles are sets with
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s satisfying axioms analogous to the
Reidemeister move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Götting ...
s used to manipulate
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
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
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
in a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
.
History
In 1943, Mituhisa Takasaki (高崎光久) introduced an algebraic structure which he called a ''Kei'' (圭), 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 Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
in the context of
finite geometry
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
. The idea was rediscovered and generalized in (unpublished) 1959 correspondence between
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
and
Gavin Wraith,
who at the time were undergraduate students at the
University of Cambridge
, mottoeng = Literal: From here, light and sacred draughts.
Non literal: From this place, we gain enlightenment and precious knowledge.
, established =
, other_name = The Chancellor, Masters and Schola ...
. It is here that the modern definitions of quandles and of racks first appear. Wraith had become interested in these structures (which he initially dubbed sequentials) while at school.
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
when one discards the multiplicative structure and considers only the
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
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 was coined),
in a 1982 paper by
Sergei Matveev (under the name
distributive groupoids
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*''Group'' with a partial func ...
)
and in a 1986 conference paper by
Egbert Brieskorn
Egbert Valentin Brieskorn (7 July 1936, in Rostock – 11 July 2013, in Bonn) was a German mathematician who introduced Brieskorn spheres and the Brieskorn–Grothendieck resolution.
Education
Brieskorn was born in 1936 as the son of a mill const ...
(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
Colin Rourke (born 1 January 1943) is a British mathematician who worked in PL topology, low-dimensional topology, differential topology, group theory, relativity and cosmology. He is an emeritus professor at the Mathematics Institute of the Un ...
and
Roger Fenn.
Racks
A rack may be defined as a set
with a binary operation
such that for every
the self-distributive law holds:
:
and for every
there exists a unique
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
such that
as
We then have
:
and thus
:
and
:
Using this idea, a rack may be equivalently defined as a set
with two binary operations
and
such that for all
#
(left self-distributive law)
#
(right self-distributive law)
#
#
It is convenient to say that the element
is acting from the left in the expression
and acting from the right in the expression
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
and
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
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s 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.
Quandles
A quandle is defined as a rack,
such that for all
:
or equivalently
:
Examples and applications
Every group gives a quandle where the operations come from conjugation:
:
In fact, every equational law satisfied by
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
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
Tame may refer to:
*Taming, the act of training wild animals
* River Tame, Greater Manchester
*River Tame, West Midlands and the Tame Valley
* Tame, Arauca, a Colombian town and municipality
* "Tame" (song), a song by the Pixies from their 1989 a ...
in
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
has a 'fundamental quandle'. To define this, one can note that the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the knot complement, or
knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot ''K'' is defined as the fundamental group of the knot complement of ''K'' in R3,
:\pi_1(\mathbb^3 \setminus K).
Other conventions co ...
, has a presentation (the
Wirtinger presentation In mathematics, especially in group theory, a Wirtinger presentation is a finite presentation where the relations are of the form wg_iw^ = g_j where w is a word in the generators, \. Wilhelm Wirtinger observed that the complements of knots in 3-sp ...
) in which the relations only involve conjugation. So, this presentation can also be used as a presentation of a quandle. The fundamental quandle is a very powerful invariant of knots. In particular, if two knots have
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
fundamental quandles then there is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
of three-dimensional Euclidean space, which may be
orientation reversing, taking one knot to the other.
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
subject to certain constraints. More sophisticated invariants of this sort can be constructed with the help of quandle
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
.
The are also important, since they can be used to compute the
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...
of a knot. Let
be a module over the ring