Biquandle

In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory of classical knots. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.
Definitions.
Biquandles and biracks have two binary operations on a set formula_1 written formula_2 and formula_3. These satisfy the following three axioms:
1. formula_4
2. formula_5
3. formula_6
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 formula_7 for formula_3 and formula_9 for formula_2 then the
three axioms above become
1. formula_11
2. formula_12
3. formula_13
If in addition the two operations are invertible, that is given formula_14 in the set formula_1 there are unique formula_16 in the set formula_1 such that formula_18 and formula_19 then the set formula_1 together with the two operations define a birack.
For example, if formula_1, with the operation formula_2, is a rack then it is a birack if we define the other operation to be the identity, formula_23.
For a birack the function formula_24 can be defined by
Then
1. formula_26 is a bijection
2. formula_27
In the second condition, formula_28 and formula_29 are defined by formula_30 and formula_31. This condition is sometimes known as the set-theoretic Yang-Baxter equation.
To see that 1. is true note that formula_32 defined by
is the inverse to
To see that 2. is true let us follow the progress of the triple formula_35 under formula_36. So
On the other hand, formula_38. Its progress under formula_39 is
Any formula_26 satisfying 1. 2. is said to be a "switch" (precursor of biquandles and biracks).
Examples of switches are the identity, the "twist" formula_42 and formula_43 where formula_2 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.
Biquandles.
A biquandle is a birack which satisfies some additional structure, as described by Nelson and Rische. The axioms of a biquandle are "minimal" in the sense that they are the weakest restrictions that can be placed on the two binary operations while making the biquandle of a virtual knot invariant under Reidemeister moves.