Biquandle

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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  X written  a^b and  a_b . These satisfy the following three axioms:

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}

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_1S_2S_1=S_2S_1S_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_1S_2S_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_2S_1S_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[edit]

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 J. Combin. 20 (1999), 663–690.