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 ${\displaystyle a^{b}}$ and ${\displaystyle a_{b}}$. These satisfy the following three axioms:

1. ${\displaystyle a^{bc_{b}}={a^{c}}^{b^{c}}}$

2. ${\displaystyle {a_{b}}_{c_{b}}={a_{c}}_{b^{c}}}$

3. ${\displaystyle {a_{b}}^{c_{b}}={a^{c}}_{b^{c}}}$

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 ${\displaystyle a*b}$ for ${\displaystyle a_{b}}$ and ${\displaystyle a**b}$ for ${\displaystyle a^{b}}$ then the three axioms above become

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

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

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

For other notations see racks and quandles.

If in addition the two operations are invertible, that is given ${\displaystyle a,b}$ in the set $X$ there are unique ${\displaystyle x,y}$ in the set $X$ such that ${\displaystyle x^{b}=a}$ and ${\displaystyle y_{b}=a}$ then the set $X$ together with the two operations define a birack.

For example if $X$, with the operation ${\displaystyle a^{b}}$, is a rack then it is a birack if we define the other operation to be the identity, ${\displaystyle a_{b}=a}$.

For a birack the function ${\displaystyle S:X^{2}\rightarrow X^{2}}$ can be defined by

${\displaystyle S(a,b_{a})=(b,a^{b}).\,}$

Then

1. $S$ is a bijection

2. ${\displaystyle S_{1}S_{2}S_{1}=S_{2}S_{1}S_{2}\,}$

In the second condition, $S_{1}$ and $S_{2}$ are defined by ${\displaystyle S_{1}(a,b,c)=(S(a,b),c)}$ and ${\displaystyle S_{2}(a,b,c)=(a,S(b,c))}$. This condition is sometimes known as the set-theoretic Yang-Baxter equation.

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

${\displaystyle S'(b,a^{b})=(a,b_{a})\,}$

is the inverse to

$S \,$

To see that 2. is true let us follow the progress of the triple ${\displaystyle (c,b_{c},a_{bc^{b}})}$ under ${\displaystyle S_{1}S_{2}S_{1}}$. So

${\displaystyle (c,b_{c},a_{bc^{b}})\to (b,c^{b},a_{bc^{b}})\to (b,a_{b},c^{ba_{b}})\to (a,b^{a},c^{ba_{b}}).}$

On the other hand, ${\displaystyle (c,b_{c},a_{bc^{b}})=(c,b_{c},a_{cb_{c}})}$. Its progress under ${\displaystyle S_{2}S_{1}S_{2}}$ is

${\displaystyle (c,b_{c},a_{cb_{c}})\to (c,a_{c},{b_{c}}^{a_{c}})\to (a,c^{a},{b_{c}}^{a_{c}})=(a,c^{a},{b^{a}}_{c^{a}})\to (a,b_{a},c_{ab_{a}})=(a,b^{a},c^{ba_{b}}).}$

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

Examples of switches are the identity, the twist ${\displaystyle T(a,b)=(b,a)}$ and ${\displaystyle S(a,b)=(b,a^{b})}$ where ${\displaystyle a^{b}}$ 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.