Biracks and biquandles

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 $$ X $$ 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 $$ a\mathbin{**}b $$ for $$ a^b $$ then the three axioms above become

1. $$ (a\mathbin{**}b)\mathbin{**}(c*b)=(a\mathbin{**}c)\mathbin{**}(b\mathbin{**}c) $$

2. $$ (a*b)*(c*b)=(a*c)*(b\mathbin{**}c) $$

3. $$ (a*b)\mathbin{**}(c*b)=(a\mathbin{**}c)*(b\mathbin{**}c) $$

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 $$ S:X^2 \rightarrow X^2 $$ can be defined by


 * $$ S(a,b_a)=(b,a^b).\,$$

Then

1. $$ S $$ is a bijection

2. $$ S_1S_2S_1=S_2S_1S_2 \, $$

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


 * $$ S'(b,a^b)=(a,b_a)\, $$

is the inverse to


 * $$ S \,$$

To see that 2. is true let us follow the progress of the triple $$ (c,b_c,a_{bc^b}) $$ under $$ S_1S_2S_1 $$. So


 * $$ (c,b_c,a_{bc^b}) \to (b,c^b,a_{bc^b}) \to (b,a_b,c^{ba_b}) \to (a, b^a, c^{ba_b}). $$

On the other hand, $$ (c,b_c,a_{bc^b}) = (c, b_c, a_{cb_c}) $$. Its progress under $$ S_2S_1S_2 $$ is


 * $$ (c, b_c, a_{cb_c}) \to (c, a_c, {b_c}^{a_c}) \to (a, c^a, {b_c}^{a_c}) = (a, c^a, {b^a}_{c^a}) \to (a, b_a, c_{ab_a}) = (a, b^a, c^{ba_b}). $$

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.

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.