多数链圈不变量在交替链圈中检测率随交叉数指数下降
On detection probabilities of link invariants
- 分析标准链圈不变量的取值分布与检测概率
- 发现不同不变量值比例和检测概率均随交叉数呈指数衰减
- 适用于琼斯多项式等经典不变量，适合拓扑学研究者
我们证明，对于许多标准链圈不变量，其在最多n个交叉的素交替链圈中，不同不变量值的比例以及检测概率均随n呈指数衰减，且具有明确的统一速率。事实上，几乎所有此类链圈都属于大小本身指数于n的不变量纤维。该现象广泛适用，尤其涵盖琼斯多项式、HOMFLYPT多项式及整数柯万诺夫同调。配套网站提供了更详细的数据显示，包括纤维大小分布、交替与非交替数据分离，以及拓扑数据分析。
原文摘要 · Abstract (English)
We prove that, for many standard link invariants, both the proportion of distinct invariant values and the detection probability among prime alternating links with at most n crossings decay exponentially in n, with an explicit universal rate. In fact, almost every such link belongs to an invariant fiber whose size is itself exponential in n. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. The companion website gives a much more detailed view of the data, including complete distributions of fiber sizes, separate alternating and non-alternating data, and topological data analysis.
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