AI 中文总结
该研究证明源于Nichols代数与Yetter-Drinfeld模的V_n纽结不变量，与由U_q(sl(2|1))-模定义的有色Links-Gould不变量一致，给出了明确的参数关系，还验证了相关标量恒等性质的猜想。
AI 中文摘要
纽结不变量V_n源于秩2的Nichols代数和4n维右Yetter-Drinfeld模，而有色Links-Gould不变量由典型U_q(sl(2|1))-模定义。我们证明这两种构造一致，仅因右模约定需考虑镜像和参数反转。证明为结构性的：首先将辫子算子T_n实现为规范超Yetter-Drinfeld编织的规范变换；接着构造相关的右-右配对双代数，证明其Hopf配对在每个根次数下都是完美的，并得到其完备通用R-矩阵；一个显式的Abel Drinfeld扭转将此双代数分解为全奇的U_ℏ(sl(2|1))根因子和交换中心因子；在该分解下，Nichols模变为典型全奇最高权模与一维中心模的张量积；最后，我们传递对偶性、所有四个定向交叉、部分转置和拧转归一化。对每个定向纽结K、所有n≥1及所有β≠0,-1，结果为V_{n,K}(Q^{2nβ+n},Q^2)=LG_K^{(n)}(Q^{-nβ},Q^{-1})=LG_{\bar{K}}^{(n)}(Q^{nβ},Q)。特别地，针对取值为自同态的V_n构造所猜想的标量恒等性质，可由对应典型模的单性推出。
英文摘要
The knot invariants \(V_n\) arise from a rank-two Nichols algebra and a \(4n\)-dimensional right Yetter--Drinfeld module, whereas the colored Links--Gould invariants are defined from typical \(U_q(\mathfrak{sl}(2|1))\)-modules. We prove that these two constructions agree, up to the mirror and parameter inversion forced by the right-module convention. The proof is structural. We first realize the braid operator \(T_n\) as a gauge transform of a canonical super Yetter--Drinfeld braiding. We then construct the relevant right--right paired double, prove that its Hopf pairing is perfect in every root degree, and obtain its completed universal \(R\)-matrix. An explicit Abelian Drinfeld twist separates this double into an all-odd \(U_\hbar(\mathfrak{sl}(2|1))\) root factor and a commutative central factor. Under this factorization, the Nichols module becomes a typical all-odd highest-weight module tensored with a one-dimensional central module. Finally, we transport duality, all four oriented crossings, partial transposes, and writhe normalization. For every oriented knot \(\mathcal K\), every \(n\geq 1\), and every \(β\neq 0,-1\), the result is \[ V_{n,\mathcal K}\!\left(Q^{2nβ+n},Q^2\right) = LG_{\mathcal K}^{(n)}\!\left(Q^{-nβ},Q^{-1}\right) = LG_{\overline{\mathcal K}}^{(n)}\!\left(Q^{nβ},Q\right). \] In particular, the scalar-identity property conjectured for the endomorphism-valued \(V_n\) construction follows from the simplicity of the corresponding typical module.
