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 $S:X^2 \rightarrow X^2$ 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.

Further reading

• [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 Journal of Combinatorics 20 (1999), 663–690.