翻訳と辞書 Words near each other ・ BiPu ・ Bipul Chettri ・ Bipul Das ・ Bipul Sharma ・ Bipunctiphorus ・ Bipunctiphorus dimorpha ・ Bipunctiphorus dissipata ・ Bipunctiphorus euctimena ・ Bipunctiphorus nigroapicalis ・ Bipunctiphorus pelzi ・ Bipyramid ・ Bipyridine ・ Biqasqas ・ Biqraqa ・ Biquadratic field ・ Biquandle ・ Biquardus ・ Biquaternion ・ Biquaternion algebra ・ Biquefarre ・ Biquinhas ・ Biquini Cavadão ・ Biquz ・ BIR ・ Bir (Hepsi album) ・ Bir al-Basha ・ Bir al-Ghanam ・ Bir al-Helou ・ Bir al-Maksur ・ Bir Ali Ben Khélifa Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Biquandle ： ウィキペディア英語版 Biquandle 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. $a^=\; ^$ 2. $\_=\; \_$ 3. $^=\; \_$ These identities appeared in 1992 in reference () 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 $S:X^2\; \backslash rightarrow\; X^2$ can be defined by : $S(a,b\_a)=(b,a^b).\backslash ,$ Then 1. $S$ is a bijection 2. $S\_1S\_2S\_1=S\_2S\_1S\_2\; \backslash ,$ 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 $S\text{'}$ defined by : $S\text{'}(b,a^b)=(a,b\_a)\backslash ,$ is the inverse to :$S\; \backslash ,$ To see that 2. is true let us follow the progress of the triple $(c,b\_c,a\_)$ under $S\_1S\_2S\_1$. So : $(c,b\_c,a\_)\; \backslash to\; (b,c^b,a\_)\; \backslash to\; (b,a\_b,c^)\; \backslash to\; (a,\; b^a,\; c^).$ On the other hand, $(c,b\_c,a\_)\; =\; (c,\; b\_c,\; a\_)$. Its progress under $S\_2S\_1S\_2$ is : $(c,\; b\_c,\; a\_)\; \backslash to\; (c,\; a\_c,\; ^)\; \backslash to\; (a,\; c^a,\; ^)\; =\; (a,\; c^a,\; \_)\; \backslash to\; (a,\; b\_a,\; c\_)\; =\; (a,\; b^a,\; c^).$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Biquandle」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.