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Biquandle

In mathematics, biquandles and biracks are generalizations of quandles and racks. Whereas the hinterland of quandles and racks is the theory of classical knots, that of the bi-versions, is the theory of virtual knots.

Biquandles and biracks have two binary operations on a set X written a and a. These satisfy the following three axioms:

1. a = a

2. a = a

3. a = a

These identities appeared in 1992 in reference [FRS] where the object was called a species.

The superscript and subscript notation is useful here because it dispenses with the need for brackets. For example if we write a * b for a and a *  * b for a then the three axioms above become

1. (a *  * b) *  * (c * b) = (a *  * c) *  * (b *  * c)

2. (a * b) * (c * b) = (a * c) * (b *  * c)

3. (a * b) *  * (c * b) = (a *  * c) * (b *  * c)

For other notations see racks and quandles.

If in addition the two operations are invertible, that is given a, b in the set X there are unique x, y in the set X such that x = a and y = a then the set X together with the two operations define a birack.

For example if X, with the operation a, is a rack then it is a birack if we define the other operation to be the identity, a = a.

For a birack the function S : X → X can be defined by

S(a, b) = (b, a). 

Then

1. S is a bijection

2. SSS = SSS

In the second condition, S and S are defined by S(a, b, c) = (S(a, b), c) and S(a, b, c) = (a, S(b, c)). This condition is sometimes known as the set-theoretic Yang-Baxter equation.

To see that 1. is true note that Sʹ defined by

Sʹ(b, a) = (a, b) 

is the inverse to


S

To see that 2. is true let us follow the progress of the triple (c, b, a) under SSS. So

(c, b, a) → (b, c, a) → (b, a, c) → (a, b, c).

On the other hand, (c, b, a) = (c, b, a). Its progress under SSS is

(c, b, a) → (c, a, b) → (a, c, b) = (a, c, b) → (a, b, c) = (a, b, c).

Any S satisfying 1. 2. is said to be a switch (precursor of biquandles and biracks).

Examples of switches are the identity, the twist T(a, b) = (b, a) and S(a, b) = (b, a) where a is the operation of a rack.

A switch will define a birack if the operations are invertible. Note that the identity switch does not do this.