An Algebraic Approach to Symmetry With Applications to Knot Theory
An Algebraic Approach to Symmetry With Applications to Knot Theory
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对称性的代数方法及其在纽结理论中的应用
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发表时间：
1979
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通讯作者：
D. E. Joyce
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作者：
D. E. Joyce
通常用来研究对象对称性的代数构造是该对象的自同构群。然而，在许多几何环境中，人们可以通过物体本身的代数结构，以更亲密的方式来解释对称性。定义一个quandle是一个集合，它有两个二元运算，（x，y）7→ x。y和（x，y）7→ x。y，满足公理Q1。X . x = x。Q2.（x . y）。y = x =（x . y）。y. Q3.（x . y）。z =（x . z）。（y . z）。调用映射S（y）将x发送到x。在y的对称性。对于对称空间的每一点y，都有空间的对称性S（y）。定义x。y = x。y是x在S（y）下的像，则对称空间成为Quandle。称之为满足x的quandle。y = x。这是一个纠缠不休的困境。卢什[1]将对称空间定义为具有对合quandle结构的流形，使得每个点y都是S（y）的孤立不动点。群G的基础集合，沿着有共轭运算，x。y = y−1xy和x。y = yxy−1形成一个quandle ConjG。此外，共轭理论可以被看作是一种困境理论，因为任何方程都可以在这个意义上。和.在ConjG中对所有群G成立，在任何Quandle中也成立。如果G的中心是平凡的，则ConjG确定G。设G是群，n ≥ 2. G的n-核是集合{（x1，x2，. . .，xn）∈ G| x1x2。. . xn = 1}沿着运算（x1，x2，. . .，xn）。(y1，y2，. . .，yn）=（y-1 n xny1，y-1 1 x1y2，. . .，y −1 n−1xn−1yn）。n-核是一个n-quandle，也就是说，每个对称都有能整除n的阶数。群G是单的当且仅当它的n-核是单quandle。设G是一个非循环单群，Q是H中的一个非平凡共轭类，Q是ConjG的一个子quandle. Q是一个简单的quandle。
The usual algebraic construction used to study the symmetries of an object is the group of automorphisms of that object. In many geometric settings, however, one may interpret the symmetries in a more intimate manner by an algebraic structure on the object itself. Define a quandle to be a set equipped with two binary operations, (x, y) 7→ x . y and (x, y) 7→ x . y, which satisfies the axioms Q1. x . x = x. Q2. (x . y) . y = x = (x . y) . y. Q3. (x . y) . z = (x . z) .(y . z). Call the map S(y) sending x to x . y the symmetry at y. To each point y of a symmetric space there is a symmetry S(y) of the space. By defining x . y = x . y to be the image of x under S(y), the symmetric space becomes a quandle. Call a quandle satisfying x . y = x . y an involutory quandle. Loos [1] has defined a symmetric space as a manifold with an involutory quandle structure such that each point y is an isolated fixed point of S(y). The underlying set of a group G along with the operations of conjugation, x . y = y−1xy and x . y = yxy−1 form a quandle ConjG. Moreover, the theory of conjugation may be regarded as the theory of quandles in the sense that any equation in . and . holding in ConjG for all groups G also holds in any quandle. If the center of G is trivial, then ConjG determines G. Let G be a group and n ≥ 2. The n-core of G is the set {(x1, x2, . . . , xn) ∈ G |x1x2 . . . xn = 1} along with the operation (x1, x2, . . . , xn) .(y1, y2, . . . , yn) = (y −1 n xny1, y −1 1 x1y2, . . . , y −1 n−1xn−1yn). The n-core is an n-quandle, that is, each symmetry has order dividing n. The group G is simple if and only if its n-core is a simple quandle. Let G be a noncyclic simple group and Q a nontrivial conjugacy class in H viewed as a subquandle of ConjG. Then Q is a simple quandle.