nLab
rack

Idea

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.

Definition

A rack may be defined as a set $R$ with a binary operation $▹$ such that for every $a,b,c\in R$ the self-distributive law holds:

$$a▹(b▹c)=(a▹b)▹(a▹c)$$a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright c)

and for every $a,b\in R$ there exists a unique $c\in R$ such that

$$a▹c=b$$a \triangleright c = b

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 $a▹c=b$ as $b◃a$. We then have

$$a▹c=b\iff c=b◃a$$a \triangleright c = b \iff c = b \triangleleft a

and thus

$$a▹(b◃a)=b$$a \triangleright (b \triangleleft a) = b

and

$$(a▹b)◃a=b$$(a \triangleright b) \triangleleft a = b

Using this idea, a rack may be equivalently defined as a set $R$ with two binary operations $▹$ and $◃$ such that for all $a,b,c\in R$:

(1)$$a▹(b▹c)=(a▹b)▹(a▹c)$$a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright c)

(2)$$(c◃b)◃a=(c◃a)◃(b◃a)$$(c \triangleleft b) \triangleleft a = (c \triangleleft a)\triangleleft (b \triangleleft a)

(3)$$(a▹b)◃a=b$$(a \triangleright b)\triangleleft a = b

(4)$$a▹(b◃a)=a$$a \triangleright (b \triangleleft a) = a

It is convenient to say that the element $a\in R$ is acting from the left in the expression $a▹b$, and acting from the right in the expression $b◃a$. 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.

Notations and Conventions

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

$$a▹b={}^{a}b$$a \triangleright b = {}^a b

and

$$b◃a=b^a$$b \triangleleft a = b^{a}

while many others write

$$b◃a=b\star a$$b \triangleleft a = b \star a

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:

$$a▹(b◃c)=(a▹b)◃(a▹c)$$a \triangleright (b \triangleleft c) = (a \triangleright b)\triangleleft (a \triangleright c)

$$(c▹b)◃a=(c◃a)▹(b◃a)$$(c \triangleright b) \triangleleft a = (c \triangleleft a)\triangleright (b \triangleleft a)

which are consequences of the definition(s) given earlier.

References

• (popular) Gavin Wraith, A personal story about knots.

• Sam Nelson, Quandle theory. link not working!

• 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.

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

• Simon Covez, The local integration of Leibniz algebras, arXiv:1011.4112; On the conjectural cohomology for groups, arXiv:1202.2269; L’intégration locale des algèbres de Leibniz, Thesis (2010), pdf