Racks and quandles

In mathematics, racks and quandles are sets with a binary operation satisfying axioms analogous to the Reidemeister moves of knot diagram manipulation.

While studied primarily in a knot-theoretic context, they can be viewed as algebraic constructions in their own right.

History

The earliest known work on racks is contained within (unpublished) 1959 correspondence between John Conway and Gavin Wraith, who at the time were undergraduate students at the University of Cambridge.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. 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 is coined), in a 1982 paper by Sergei Matveev (under the name distributive groupoids) and in a 1986 conference paper by Egbert Brieskorn (where they are called automorphic sets).

Quandles

A quandle is defined as a set $mathrm\{Q\}$ with a binary operation $star$ such that:$forall\; a,\; b,\; c\; in\; mathrm\{Q\}$
#$astar\; a=a$
(the operation is reflexive)
#$exists\; !\; x\; in\; mathrm\{Q\}\; :\; x\; star\; a\; =\; b$
(acting on the right by a given element is a bijection)
#$(a\; star\; b)\; star\; c\; =\; (a\; star\; c)\; star\; (b\; star\; c)$
(the operation is right-distributive over itself)

It is convenient to consider in $a\; star\; b$ that $b$ is acting from the right on $a$. The second quandle axiom says that this action is a bijection for each $b$.

Every group gives a quandle where the operation $star$ comes from conjugation:

: $a\; star\; b\; =\; b^\{-1\}\; a\; b$

In fact, every equation satisfied by conjugation in a group follows from the three quandle axioms. So, one can think of a quandle as what's 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. 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 fundamental quandles then there is a homeomorphism of space which may be orientation reversing taking one knot to the other.

A quandle $mathrm\{Q\}$ is said to be 'involutory' if $forall\; a,\; b\; in\; mathrm\{Q\}$

: $(a\; star\; b)\; star\; b\; =\; a$

This equation makes the second quandle axiom redundant, since it guarantees that acting by $b$ on the right is its own inverse.

Any symmetric space gives an involutory quandle, where $a\; star\; b$ is the result of 'reflecting $a$ through $b$'.

Racks

A rack is more general than a quandle: it satisfies only the two last properties. Thus, a rack is defined as a set $mathrm\{R\},!$ paired with a binary operation $star;$ such that:$forall;\; a,\; b,\; c\; in;\; mathrm\{R\},!$
#$exists\; !\; x\; in\; mathrm\{R\},!\; :\; x\; star;\; a\; =\; b$
(acting on the right by a given element is a bijection)
#$(a\; star;\; b)\; star;\; c\; =\; (a\; star;\; c)star;\; (b\; star;\; c)$
(the operation is right-distributive over itself)

The use of $star$ is by no means universal: some authors use exponential notation to reflect the inherent asymmetry of the operation.

An alternative but equivalent definition of a rack is that it is a set with a binary operation in which multiplication on the right is an automorphism.

Whereas 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.

See also birack and biquandle.

External links

* [http://www.wra1th.plus.com/gcw/rants/math/Rack.html A Personal Story about Knots] by Gavin Wraith

References

* John Conway, Gavin Wraith, unpublished correspondence (1959)
* David Joyce, "A classifying invariant of knots: the knot quandle", Journal of Pure and Applied Algebra 23 (1982) 37–65
* Sergei Matveev, "Distributive groupoids in knot theory", Matematicheskiui Sbornik 119 (1982) 78–88, 160
* Egbert Brieskorn, "Automorphic sets and singularities", in "Braids (Santa Cruz, CA, 1986)", Contemporary Mathematics 78 (1988) 45–115
* Roger Fenn, Colin Rourke, [http://www.maths.warwick.ac.uk/~cpr/ftp/racks.ps "Racks and links in codimension 2"] , Journal of Knot Theory and its Ramifications 1 (1992) 343–406