Biquandle

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[edit] Biracks

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 ab and ab. 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 ab and a * * b for ab 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 .

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 xb = a and yb = a then the set X together with the two operations define a birack.

For example if X, with the operation ab, is a rack then it is a birack if we define the other operation to be the identity, ab = a.

For a birack the function S:X2 − > X2 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, S1 and S2 are defined by S1(a,b,c) = (S(a,b),c) and S2(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 S1S2S1. 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 S2S1S2 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,ab) where ab 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.

[edit] Biquandles

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.

[edit] Linear biquandles

In Preparation

[edit] Application to virtual links and braids

In Preparation

[edit] Birack homology

In Preparation

[edit] References

  • [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.