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# Biquandle

## Biracks

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. $a^\left\{b\left\{c_b\right\}\right\}= \left\{a^c\right\}^\left\{b^c\right\}$

2. $\left\{a_b\right\}_\left\{c_b\right\}= \left\{a_c\right\}_\left\{b^c\right\}$

3. $\left\{a_b\right\}^\left\{c_b\right\}= \left\{a^c\right\}_\left\{b^c\right\}$

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. $\left(a**b\right)**\left(c*b\right)=\left(a**c\right)**\left(b**c\right)$

2. $\left(a*b\right)*\left(c*b\right)=\left(a*c\right)*\left(b**c\right)$

3. $\left(a*b\right)**\left(c*b\right)=\left(a**c\right)*\left(b**c\right)$

For other notations see .

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

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

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 $S_1\left(a,b,c\right)=\left(S\left(a,b\right),c\right)$ and $S_2\left(a,b,c\right)=\left(a,S\left(b,c\right)\right)$. This condition is sometimes known as the set-theoretic Yang-Baxter equation.

To see that 1. is true note that $S\text{'}$ defined by

$S\text{'}\left(b,a^b\right)=\left(a,b_a\right),$

is the inverse to

$S ,$

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

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

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

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

Any $S$ satisfying 1. 2. is said to be a switch (precursor of biquandles and biracks).

Examples of switches are the identity, the twist $T\left(a,b\right)=\left(b,a\right)$ and $S\left(a,b\right)=\left(b,a^b\right)$ 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. It should be noted that 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.

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## References

• [FJK] Roger Fenn, Mercedes Jordan-Santana, Louis Kauffman Biquandles and Virtual Links, Topology and its Applications, 145 (2004) 157-175
• [FRS] Roger Fenn, Colin Rourke, Brian Sanderson An Introduction to Species and the Rack Space, in Topics in Knot Theory (1992), Kluwer 33-55
• [K] L. H. Kauffman, Virtual Knot Theory, European J. Combin. 20 (1999), 663--690.
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