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 b and a b . These satisfy the following three axioms:

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

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 b and a ∗ ∗ b for a b 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 b = a and y b = a then the set X together with the two operations define a birack.

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

For a birack the function S : X 2 → X 2 can be defined by

Then

1. S is a bijection

2. S 1 S 2 S 1 = S 2 S 1 S 2

In the second condition, S 1 and S 2 are defined by S 1 ( a , b , c ) = ( S ( a , b ) , c ) and S 2 ( 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 S ′ defined by

is the inverse to

To see that 2. is true let us follow the progress of the triple ( c , b c , a b c b ) under S 1 S 2 S 1 . So

On the other hand, ( c , b c , a b c b ) = ( c , b c , a c b c ) . Its progress under S 2 S 1 S 2 is

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 b ) where a b 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.