发表机构
Instituto Tecnológico Autónomo de México (ITAM); Colegio de Matemáticas Bourbaki; Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV-IPN); North Carolina State University; Tecnológico de Monterrey（墨西哥自治理工学院; 布尔巴基数学学院; 国立理工学院高级研究中心; 北卡罗来纳州立大学; 蒙特雷科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明在随机图模型下，归一化Alexander多项式区分纽结对但碰撞熵为$O(n^{-1/2})$，行列式碰撞频率低，并给出碰撞概率与生日尺度分析。
AI 中文摘要
纽结不变量是等价纽结共享的指纹：不等的值证明不等价，而相等的值可能隐藏不同的纽结。在两种有根随机图模型下，我们证明了不完全归一化Alexander多项式能区分随机对，但在增长的数据集中会发生碰撞。若$D_n$和$D_n'$是独立的$n$交叉图，则$\Pr\{\Delta_{K(D_n)}=\Delta_{K(D_n')}\}=O(n^{-1/2})$，且每个固定的归一化Alexander多项式都是指数级稀有的。以指数级高的概率，图的阴影包含线性多个不相交的开三叶结槽。在给定阴影和外部交叉符号的条件下，它们的指示变量是独立的Bernoulli$(1/4)$变量，其和是$3$-adic行列式赋值中的二项坐标。因此，$\sup_{a\geq 1}\Pr\{\det K(D_n)=a\}=O(n^{-1/2})$。对于离散不变量$I$，令$\alpha_n(I)=\Pr\{I(D_n)=I(D_n')\}$。其碰撞熵为$H_2(I(D_n))=-\log\alpha_n(I)$。大小为$M$的独立样本平均有$\binom{M}{2}\alpha_n(I)$个碰撞对，生日尺度为$\alpha_n(I)^{-1/2}$。若$M_n^2\alpha_n(I)\to\infty$，则即使$\alpha_n(I)\to0$，出现重复值的概率也趋于1。对于行列式，$M=o(n^{1/4})$足以高概率避免碰撞；匹配的下界仍未解决。我们还给出了固定普查的精确有限总体公式，计算了不变量级联的期望成本，并将碰撞概率与完整等价过程的调用频率联系起来。在平衡的成对分类基准上，归一化Alexander规则具有平衡准确率$1-O(n^{-1/2})$，但无法区分同一纤维中的不等价对。
英文摘要
A knot invariant is a fingerprint shared by equivalent knots: unequal values certify inequivalence, while equal values may conceal different knots. Under two rooted random-diagram models, we prove that the incomplete normalized Alexander polynomial separates random pairs but collides in a growing database. If $D_n$ and $D_n'$ are independent $n$-crossing diagrams, then $\Pr\{Δ_{K(D_n)}=Δ_{K(D_n')}\}=O(n^{-1/2})$, and every fixed normalized Alexander polynomial is exponentially rare. With exponentially high probability, the diagram shadow contains linearly many disjoint opened-trefoil slots. Conditional on the shadow and exterior crossing signs, their indicators are independent Bernoulli$(1/4)$ variables whose sum is a binomial coordinate in the $3$-adic determinant valuation. Consequently, $\sup_{a\geq 1}\Pr\{\det K(D_n)=a\}=O(n^{-1/2})$. For a discrete invariant $I$, let $α_n(I)=\Pr\{I(D_n)=I(D_n')\}$. Its collision entropy is $H_2(I(D_n))=-\logα_n(I)$. An independent sample of size $M$ has $\binom{M}{2}α_n(I)$ colliding pairs on average and birthday scale $α_n(I)^{-1/2}$. If $M_n^2α_n(I)\to\infty$, a repeated value occurs with probability tending to one even when $α_n(I)\to0$. For the determinant, $M=o(n^{1/4})$ suffices for collision freedom with high probability; a matching lower bound is open. We also give exact finite-population formulas for fixed censuses, calculate the expected cost of invariant cascades, and relate collision probability to the frequency of calls to a complete equivalence procedure. On a balanced pair-classification benchmark, the normalized-Alexander rule has balanced accuracy $1-O(n^{-1/2})$ but cannot distinguish inequivalent pairs in one fiber.
Comments12 pages, 2 figures. This arXiv version contains the main article only
