The \(V_n\) Invariants as Colored Links--Gould Invariants:A Root--Center Approach

Abstract

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 \(\beta\neq 0,-1\), the result is \[ V_{n,\mathcal K}\!\left(Q^{2n\beta+n},Q^2\right) = LG_{\mathcal K}^{(n)}\!\left(Q^{-n\beta},Q^{-1}\right) = LG_{\overline{\mathcal K}}^{(n)}\!\left(Q^{n\beta},Q\right). \] In particular, the scalar-identity property conjectured for the endomorphism-valued \(V_n\) construction follows from the simplicity of the corresponding typical module.