Racks and quandles

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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.

1 History
2 Quandles
3 Racks
4 External links
5 References

[edit] 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The earliest known work on racks is contained within (unpublished) 1959 correspondence between John Conway and Gavin Wraith, who at the time were undergraduate students at the University of Cambridge.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).

[edit] Quandles

A quandle is defined as a set $Q$ paired with a binary operation $\star$ such that: $\forall a, b, c \in \mathrm{Q}$

$a\star a=a$
(the operation is reflexive)
$\exists\ ! x \in \mathrm{Q} : x \star a = b$
(multiplication on the right by a given element is a permutation)
$(a \star b) \star c = (a \star c) \star (b \star c)$
(the operation is right distributive)

The symbol $\star$ does not denote multiplication; the quandle operation can be defined in any matter on a set so long as it adheres to the three properties of a quandle. It is convenient to consider in $a \star b$ that $b$ is acting from the right on $a$ . So the second condition implies that this action is a bijection for each $b$ .

Every tame knot in three dimensional euclidean space has a fundamental quandle. 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.

[edit] 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: $\forall\; a, b, c \in\; \mathrm{R}\,\!$

$\exists\ ! x \in\ \mathrm{R}\,\! : x\ \star\; a = b$
(multiplication on the right by a given element is a permutation)
$(a\ \star\; b)\ \star\; c = (a\ \star\; c)\star\; (b\ \star\; c)$
(the operation is right distributive)

As with quandles, the symbol $\star\;$ does not denote an ordinary multiplication; the rack operation can be defined in any matter on a set so long as it adheres to the two properties of a rack. The use of $\star$ is by no means universal: some authors use exponential notation to reflect the inherent asymmetry of the operation.

An alternative definition of a rack is that it is a set with a binary operation in which multiplication on one side 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.

[edit] External links

Quandle theory
A Personal Story about Knots by Gavin Wraith

[edit] 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, Racks and links in codimension 2, Journal of Knot Theory and its Ramifications 1 (1992) 343–406