A version 2 arXiv preprint dated 1 September 2026 reports an exact algebraic match between two constructions of knot invariants: the Nichols-algebra V_n invariants and the colored Links-Gould invariants. The paper says the match comes with a mirror and parameter inversion dictated by its use of a right-module convention.
The result is presented as a structural proof. Instead of comparing two large matrices entry by entry, the paper builds a chain of identifications between the algebraic objects behind the two invariant families.
At the center is a basis-defined space called Y_n, used in the V_n construction. Its reported dimension is 4n, where n is the color index. The endomorphism-valued construction is defined on this space.
The machinery behind the match
One early step establishes a continuous, even Hopf pairing between the positive and negative Hopf superalgebras. The pairing is reported to be perfect in every root degree, and it produces a completed universal R-matrix for the right-right paired double.
The paper says the perfectness step uses PBW normal forms and explicit root-degree pairing matrices. It relies on those calculations to establish the pairing degree by degree.
An explicit Abelian Drinfeld twist then separates the double into two pieces: an all-odd quantum sl(2|1) root factor and a commutative central factor. This factorization is the paper's main structural bridge from the Nichols-algebra side to the quantum-superalgebra side.
Under that factorization, the Nichols module becomes a typical all-odd highest-weight module tensored with a one-dimensional central module. The module-side description therefore separates the quantum sl(2|1) component from the central line.
Carrying the result to knots
The structural identification is carried through the braid data used for oriented knots. The equivalence extends coherently to dual objects, evaluation and coevaluation, all four oriented crossing types, and zero-framing normalization.
The central factor contributes a scalar at each positive crossing and the inverse scalar at each negative crossing. The paper says those contributions cancel on balanced normal diagrams and under writhe normalization, leaving the normalized invariant aligned with the root-side construction.
The other headline result concerns the long-knot invariant that takes values in endomorphisms of Y_n. It is a scalar multiple of the identity on that space, so it acts by the same scalar across Y_n rather than changing from one direction in the space to another.
A result with defined boundaries
The paper's conclusion is narrower than an equivalence of all the surrounding algebra. It compares normalized oriented-knot invariants, uses a formal parameter ring, and excludes beta = 0 and beta = -1. It also does not claim that the entire module categories of the completed Hopf superalgebras are equivalent.
The equality is a theorem-level statement inside the paper's specified formal and completed setting. Its interpretation depends on the stated right-module, mirror, parameter and writhe-normalization conventions.
Publication notes
The supplied publication record identifies the work as an arXiv preprint, version 2, dated 1 September 2026. The acknowledgments name support from an adviser, family, friends and a partner, but do not report a grant or sponsor.
The acknowledgments also disclose the use of large language models to generate and refine portions of the text, followed by substantial human intervention.
Paper data and sources
Original title: The Invariants as Colored Links--Gould Invariants:A Root--Center Approach
Authors: Jiuhe Liu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text