The Vn Invariants as Colored Links--Gould Invariants:A Root--Center Approach
Abstract
The knot invariants Vn 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 Uq(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 Tn 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 Uℏ(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 𝒦, every n≥ 1, and every β≠ 0,−1, the result is Vn,𝒦 (Q2nβ+n,Q2) = LG𝒦(n) (Q−nβ,Q−1) = LG𝒦̅(n) (Qnβ,Q). In particular, the scalar-identity property conjectured for the endomorphism-valued Vn construction follows from the simplicity of the corresponding typical module.
Related
- The Colored Jones Function does not determine any colored Links--Gould polynomial
- The colored Jones polynomial does not determine the Links--Gould polynomial
- The Links--Gould polynomial of an alternating link is alternating
- The top coefficient of the Links-Gould invariant is Murasugi-sum multiplicative
- A lower bound for the genus of a knot using the Links–Gould invariant
BibTeX
@misc{liu2026invariants,
title = {{The \(V_n\) Invariants as Colored Links--Gould Invariants:A Root--Center Approach}},
author = {Jiuhe Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.28720},
eprint = {2608.28720},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28720},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The V_n Invariants as Colored Links--Gould Invariants:A Root--Center Approach
AU - Jiuhe Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.28720
UR - https://arxiv.org/abs/2608.28720
AB - 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(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_ℏ(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 𝒦, every n≥ 1, and every β≠ 0,−1, the result is V_n,𝒦 (Q^2nβ+n,Q^2) = LG_𝒦^(n) (Q^−nβ,Q^−1) = LG_𝒦̅^(n) (Q^nβ,Q). In particular, the scalar-identity property conjectured for the endomorphism-valued V_n construction follows from the simplicity of the corresponding typical module.
ER -