Conway algebras and skein equivalence of links
Conway algebras and skein equivalence of links
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DOI：
10.1090/s0002-9939-1987-0894448-2
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发表时间：
1987-04
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通讯作者：
J. Przytycki;Pawel Traczyk
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文献类型：
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作者：
J. Przytycki;Pawel Traczyk
我们考虑一类链对，它们不是绞等价的，但在每个康威代数中具有相同的不变量。1. Conway代数首先，我们将回顾的概念康威代数介绍[PT]。定义1.1一个康威代数是一个代数A具有一个0参数操作序列a1，a2，.以及两个2参数运算I和 *，它们满足以下条件：anIan+1 = anC2。an * an+1 = an（C1和C2是初始条件属性），C3。（alb）I（cld）=（alc）I（bld），C4。(aIb)* （cId）=（a * c）I（B * d）（C3、C4和C5是转座性质），C5.（a*B）*（c*d）=（a*c）*（B*d），C6. (alb)* B = a，C7。（a*B）lb=a。如[PT]所示，每个康威代数产生一个不变的链接，这是常数的绞等价类（绞不变）。它由以下条件唯一确定：AT_i = an（初始关系），AL+ = AL_ JALo和AL_ = AL+ * ALo（Conway关系）。这里Tn表示n个分量的平凡链路，而L+、L-和LO是除了靠近一个交叉点之外完全相同的定向链路的图（见图1.1）。
We consider a class of pairs of links which are not skein equivalent but have the same invariant in every Conway algebra. 1. Conway algebras. We will first recall the notion of Conway algebra as introduced in [PT]. DEFINITION 1.1. A Conway algebra is an algebra A with a sequence of 0argument operations a1, a2, ... and two 2-argument operations I and *, which satisfy the following conditions: Ci. anIan+1 = ani C2. an * an+1 = an (Cl and C2 are initial conditions properties), C3. (alb)I(cld) = (alc)I(bld), C4. (aIb) * (cId) = (a * c)I(b * d) (C3, C4 and C5 are transposition properties), C5. (a*b) * (c*d) = (a*c) * (b*d), C6. (alb) * b = a, C7. (a*b)lb=a. As shown in [PT] every Conway algebra yields an invariant of links which is constant on skein equivalence classes (skein invariant). It is uniquely determined by the following conditions: AT, = an (initial relations), AL+ = AL_ JALo and AL_ = AL+ * ALO (Conway relations). Here Tn denotes a trivial link of n components and L+, L-, and LO are diagrams of oriented links identical except near one crossing point (see Figure 1.1).