具有多项式状态数的环的正则合痕的状态和不变量
A state sum invariant for regular isotopy of links having a polynomial number of states
摘要：
状态和定期同痕不变量的链接，我在这项工作中介绍的是一个推广的琼斯多项式。所以它区分了任何一对可以被琼斯区分的链接。这个新的不变量，记为{ \em VSE-invariant}严格强于琼斯的：我检测到一对链接，这对链接不被琼斯区分，但被新的不变量区分。完整的VSE不变量具有$3^n$状态。然而，有一些有用的特殊化，它由整数k参数化，具有$O(n^k)=\sum_{\ell=0}^k {n \choose \ell} 2^\ell $状态。通过VSE-不变量区分的具有更多交叉对的链路具有20个交叉。足以区分的特化对应于k=2并且只有801个状态，与相同链路的琼斯多项式的$2^{20} = 1,048,576$状态相反。它的完整VSE不变式具有$3^{20} = 3,486,784,401$状态。当交叉点的数量使得琼斯多项式的计算不可能时，VSE不变式是琼斯多项式的一个很好的替代方案。例如，对于$k=2$，具有$n=500$交叉的链路的VSE不变量的专门化可以在几分钟内计算，因为它只有$2 n^2+1 = 500,001$状态。
英文摘要：
The state sum regular isotopy invariant of links which I introduce in this work is a generalization of the Jones Polynomial. So it distinguishes any pair of links which are distinguishable by Jones'. This new invariant, denoted {\em VSE-invariant} is strictly stronger than Jones': I detected a pair of links which are not distinguished by Jones' but are distinguished by the new invariant. The full VSE-invariant has $3^n$ states. However, there are useful specializations of it parametrized by an integer k, having $O(n^k)=\sum_{\ell=0}^k {n \choose \ell} 2^\ell $ states. The link with more crossings of the pair which was distinguished by the VSE-invariant has 20 crossings. The specialization which is enough to distinguish corresponds to k=2 and has only 801 states, as opposed to the $2^{20} = 1,048,576$ states of the Jones polynomial of the same link. The full VSE-invariant of it has $3^{20} = 3,486,784,401$ states. The VSE-invariant is a good alternative for the Jones polynomial when the number of crossings makes the computation of this polynomial impossible. For instance, for $k=2$ the specialization of the VSE-invariant of a link with $n=500$ crossings can be computed in a few minutes, since it has only $2 n^2+1 = 500,001$ states.
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