The Jones polynomial of ribbon links
The Jones polynomial of ribbon links
复制标题
带状链接的琼斯多项式
DOI：
--
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
Michael Eisermann
中科院分区：
文献类型：
--
作者：
Michael Eisermann
对于每一个有n个分量的带状连杆L，我们证明了琼斯多项式V(L)可以被平凡连杆的多项式V（O^n）整除。这个完整性性质允许我们定义一个广义行列式det V(L):= [V(L)/V(O^n)]＿(t=-1)，由此我们得到了与Arf不变量类似的同余：每个带链L = （K＿1，…，K＿n）满足det V(L) = det（K＿1）…det（K＿n）模32，其中det V(L) = 1模8。这些结果促使我们研究幂级数展开式V(L) = \sum ＿k=0{^ }\infty d＿k(L) h^k在t=-1而不是通常的t=1时的展开式。我们得到了一组链路不变量d＿k(L)，从从Seifert曲面S生成L得到的链路行行式d＿0(L) = det(L)开始。不变量d＿k(L)对于L的交叉变化不是有限型的，但对于S的带交叉变化却是有限型的。这一发现是有限型曲面不变量理论的起点，它保证了量子不变量与Seifert曲面理论的协调。或者更普遍的带状表面。
For every n-component ribbon link L we prove that the Jones polynomial V(L) is divisible by the polynomial V(O^n) of the trivial link. This integrality property allows us to define a generalized determinant det V(L) := [V(L)/V(O^n)]_(t=-1), for which we derive congruences reminiscent of the Arf invariant: every ribbon link L = (K_1,...,K_n) satisfies det V(L) = det(K_1) ... det(K_n) modulo 32, whence in particular det V(L) = 1 modulo 8. These results motivate to study the power series expansion V(L) = \sum_{k=0}^\infty d_k(L) h^k at t=-1, instead of t=1 as usual. We obtain a family of link invariants d_k(L), starting with the link determinant d_0(L) = det(L) obtained from a Seifert surface S spanning L. The invariants d_k(L) are not of finite type with respect to crossing changes of L, but they turn out to be of finite type with respect to band crossing changes of S. This discovery is the starting point of a theory of surface invariants of finite type, which promises to reconcile quantum invariants with the theory of Seifert surfaces, or more generally ribbon surfaces.