Biracks and biquandles

In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory of classical knots. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.

Definitions

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Biquandles and biracks have two binary operations on a set   written   and  . These satisfy the following three axioms:

1.  

2.  

3.  

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

1.  

2.  

3.  

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

 

Then

1.   is a bijection

2.  

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.

Biquandles

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A biquandle is a birack which satisfies some additional structure, as described by Nelson and Rische.[1] 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

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Birack homology

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References

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  1. ^ Nelson, Sam; Rische, Jacquelyn L. (2008). "On bilinear biquandles". Colloquium Mathematicum. 112 (2): 279–289. arXiv:0708.1951. doi:10.4064/cm112-2-5.

Further reading

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