Biracks and biquandles

In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. In the theory of virtual knots, biquandles are analagous to quandles 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

A birack is a set, two right-invertible operations and a bijection such that for all,
1. and.
2. The map defined by is invertible.
3. The exchange laws

If is the identity map, is called a biquandle.
Biracks and biquandles were first introduced by Roger Fenn, Mercedes Jordan-Santana and Louis Kauffman in 2004.
Note that the three conditions above correspond directly to the three Reidemeister moves in knot therory, showing the close connections between knots and biracks.

Examples

Let be a set with two bijection that commute. Then the constant action birack is defined by and and.
Any rack is a birack with and for all. Note that if we insert these operations into the conditions in the definition above, we regain the exact definition of a rack.
Let be a commutative ring with identity and an -module. Then the Alexander biquandle is defined as and. For this is a quandle.