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 written and . These satisfy the following three axioms:
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 for and for then the
three axioms above become
For other notations see .
If in addition the two operations are invertible, that is given in the set there are unique in the set such that and then the set together with the two operations define a birack.
For example if , with the operation , is a rack then it is a birack if we define the other operation to be the identity, .
For a birack the function can be defined by
1. is a bijection
In the second condition, and are defined by and . This condition is sometimes known as the set-theoretic Yang-Baxter equation.
To see that 1. is true note that defined by
is the inverse to
To see that 2. is true let us follow the progress of the triple under . So
On the other hand, . Its progress under is
Any satisfying 1. 2. is said to be a switch (precursor of biquandles and biracks).
Examples of switches are the identity, the twist and where 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.
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.
Application to virtual links and braids
- [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.