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. $a^\{b\{c\_b\}\}=\; \{a^c\}^\{b^c\}$

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 .

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

$S(a,b\_a)=(b,a^b).,$

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 $S\text{'}$ defined by

$S\text{'}(b,a^b)=(a,b\_a),$

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. 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.

Linear biquandles

In Preparation

Application to virtual links and braids

In Preparation

Birack homology

In Preparation

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.