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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 racks and quandles.

If in addition the two operations are invertible, that is given in the set there are unique[disambiguation needed] 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. 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[edit]

Application to virtual links and braids[edit]

Birack homology[edit]

Further reading[edit]

  • [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.