Biquandle
Encyclopedia
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 knot
s.
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.
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
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, biquandles and biracks are generalizations of quandles and racks. Whereas the hinterland of quandles and racks is the theory of classical knots
Knot
A knot is a method of fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or several segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object—the "load"...
, that of the bi-versions, is the theory of virtual knot
Virtual knot
In knot theory, a virtual knot is a generalization of the classical idea of knots in several ways that are all equivalent, introduced by .-Overview:...
s.
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 thethree axioms above become
1.
2.
3.
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 identityIdentity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
,
.For a birack the function
can be defined by
Then
1.
is a bijectionBijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
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
Identity (mathematics)
In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
, 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.