Latimer–MacDuffee theorem

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. $a^{bc_{b}}={a^{c}}^{b^{c}}$

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

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 $S_{1}S_{2}S_{1}$. 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 $S_{2}S_{1}S_{2}$ 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.

Linear biquandles

In Preparation

Application to virtual links and braids

In Preparation

Birack homology

In Preparation

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 J. Combin. 20 (1999), 663–690.