[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/biracks-and-biquandles-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/biracks-and-biquandles-wikipedia\/","headline":"Biracks and biquandles – Wikipedia","name":"Biracks and biquandles – Wikipedia","description":"before-content-x4 Special ordered sets after-content-x4 In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks.","datePublished":"2017-01-29","dateModified":"2017-01-29","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/68baa052181f707c662844a465bfeeb135e82bab","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/biracks-and-biquandles-wikipedia\/","wordCount":7213,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Special ordered sets       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In 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.Table of Contents       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Definitions[edit]Biquandles[edit]Linear biquandles[edit]Application to virtual links and braids[edit]Birack homology[edit]Further reading[edit]Definitions[edit]Biquandles and biracks have two binary operations on a set X{displaystyle X} written ab{displaystyle a^{b}} and ab{displaystyle a_{b}}. These satisfy the following three axioms:       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x41.  (ab)cb=acbc{displaystyle (a^{b})^{c_{b}}={a^{c}}^{b^{c}}}2.  abcb=acbc{displaystyle {a_{b}}_{c_{b}}={a_{c}}_{b^{c}}}3.  abcb=acbc{displaystyle {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\u2217b{displaystyle a*b} for ab{displaystyle a_{b}} and a\u2217\u2217b{displaystyle amathbin {**} b} for ab{displaystyle a^{b}} then thethree axioms above become1. (a\u2217\u2217b)\u2217\u2217(c\u2217b)=(a\u2217\u2217c)\u2217\u2217(b\u2217\u2217c){displaystyle (amathbin {**} b)mathbin {**} (c*b)=(amathbin {**} c)mathbin {**} (bmathbin {**} c)}2. (a\u2217b)\u2217(c\u2217b)=(a\u2217c)\u2217(b\u2217\u2217c){displaystyle (a*b)*(c*b)=(a*c)*(bmathbin {**} c)}3. (a\u2217b)\u2217\u2217(c\u2217b)=(a\u2217\u2217c)\u2217(b\u2217\u2217c){displaystyle (a*b)mathbin {**} (c*b)=(amathbin {**} c)*(bmathbin {**} c)}If in addition the two operations are invertible, that is given a,b{displaystyle a,b} in the set X{displaystyle X} there are unique x,y{displaystyle x,y} in the set X{displaystyle X} such that xb=a{displaystyle x^{b}=a} and yb=a{displaystyle y_{b}=a} then the set X{displaystyle X} together with the two operations define a birack.For example, if X{displaystyle X}, with the operation ab{displaystyle a^{b}}, is a rack then it is a birack if we define the other operation to be the identity,  ab=a{displaystyle a_{b}=a}.For a birack the function S:X2\u2192X2{displaystyle S:X^{2}rightarrow X^{2}} can be defined byS(a,ba)=(b,ab).{displaystyle S(a,b_{a})=(b,a^{b}).,}Then1. S{displaystyle S} is a bijection2. S1S2S1=S2S1S2{displaystyle S_{1}S_{2}S_{1}=S_{2}S_{1}S_{2},}In the second condition,  S1{displaystyle S_{1}} and S2{displaystyle S_{2}} are defined by S1(a,b,c)=(S(a,b),c){displaystyle S_{1}(a,b,c)=(S(a,b),c)} and S2(a,b,c)=(a,S(b,c)){displaystyle 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\u2032{displaystyle S’} defined byS\u2032(b,ab)=(a,ba){displaystyle S'(b,a^{b})=(a,b_{a}),}is the inverse toS{displaystyle S,}To see that 2. is true let us follow the progress of the triple (c,bc,abcb){displaystyle (c,b_{c},a_{bc^{b}})} under S1S2S1{displaystyle S_{1}S_{2}S_{1}}. So(c,bc,abcb)\u2192(b,cb,abcb)\u2192(b,ab,cbab)\u2192(a,ba,cbab).{displaystyle (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,bc,abcb)=(c,bc,acbc){displaystyle (c,b_{c},a_{bc^{b}})=(c,b_{c},a_{cb_{c}})}. Its progress under S2S1S2{displaystyle S_{2}S_{1}S_{2}} is(c,bc,acbc)\u2192(c,ac,bcac)\u2192(a,ca,bcac)=(a,ca,baca)\u2192(a,ba,caba)=(a,ba,cbab).{displaystyle (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{displaystyle 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){displaystyle T(a,b)=(b,a)} and S(a,b)=(b,ab){displaystyle S(a,b)=(b,a^{b})} where ab{displaystyle 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]This section is empty.  You can help by adding to it.  (November 2014)Application to virtual links and braids[edit]This section is empty.  You can help by adding to it.  (November 2014)Birack homology[edit]This section is empty.  You can help by adding to it.  (November 2014)Further reading[edit][FJK] Roger Fenn, Mercedes Jordan-Santana, Louis Kauffman Biquandles and Virtual Links, Topology and its Applications, 145 (2004) 157\u2013175[FRS] Roger Fenn, Colin Rourke, Brian Sanderson An Introduction to Species and the Rack Space in Topics in Knot Theory (1992), Kluwer 33\u201355[K] L. H. Kauffman, Virtual Knot Theory, European Journal of Combinatorics 20 (1999), 663\u2013690.     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