Biquandle
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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
knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
s. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.


Definitions

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)^= ^ 2. _= _ 3. ^= _ These identities appeared in 1992 in reference RSwhere 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\mathbinb for a^b then the three axioms above become 1. (a\mathbinb)\mathbin(c*b)=(a\mathbinc)\mathbin(b\mathbinc) 2. (a*b)*(c*b)=(a*c)*(b\mathbinc) 3. (a*b)\mathbin(c*b)=(a\mathbinc)*(b\mathbinc) If in addition the two operations are
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, 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 Rack or racks may refer to: Storage and installation * Amp rack, short for amplifier rack, a piece of furniture in which amplifiers are mounted * Bicycle rack, a frame for storing bicycles when not in use * Bustle rack, a type of storage bi ...
then it is a birack if we define the other operation to be the
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
, 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 Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
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_) under S_1S_2S_1 . So : (c,b_c,a_) \to (b,c^b,a_) \to (b,a_b,c^) \to (a, b^a, c^). On the other hand, (c,b_c,a_) = (c, b_c, a_) . Its progress under S_2S_1S_2 is : (c, b_c, a_) \to (c, a_c, ^) \to (a, c^a, ^) = (a, c^a, _) \to (a, b_a, c_) = (a, b^a, c^). 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

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


Application to virtual links and braids


Birack homology


References


Further reading

* * * {{cite journal , last1=Kauffman , first1=Louis H. , title=Virtual Knot Theory , journal=
European Journal of Combinatorics European, or Europeans, or Europeneans, may refer to: In general * ''European'', an adjective referring to something of, from, or related to Europe ** Ethnic groups in Europe ** Demographics of Europe ** European cuisine, the cuisines of Europe ...
, volume=20 , issue=7 , date=1999 , pages=663–690 , doi=10.1006/eujc.1999.0314 , doi-access=free Knot theory Algebraic structures Ordered algebraic structures Non-associative algebra