What is "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.