A mathematical preprint describes a way to extract more information from links than their multivariable Alexander polynomial alone can provide. In selected two- and three-component comparisons involving representatives with at most 10 crossings, the authors exhibited three pairs explicitly and listed 14 more in which the polynomials matched up to Laurent units, or invertible monomial factors, but the second Alexander–Gröbner invariant, called AG2, was different.
The result is not a claim about every link. The calculations concern chosen oriented and ordered representatives, so the reported distinctions show what this construction can detect within the stated collection rather than establishing a universal separation rule.
Turning ideals into a fixed fingerprint
The construction has two linked steps. It begins with the kth Alexander Fitting ideal and takes its polynomial contraction, then computes a reduced Gröbner basis. For a general reader, that basis can be understood here as a finite, standardized representative of the resulting polynomial ideal. Once the coefficient field, ordered basis and monomial order are fixed, the paper says this representative is unique; the resulting objects are its Alexander–Gröbner invariants, AGk.
The fixed choices are part of the definition, not incidental details. Equivalent oriented ordered links have the same Alexander–Gröbner invariants, while treating links as unordered produces the corresponding families after component relabeling. At the same time, the reduced bases depend on the chosen coefficient field, ordered basis, orientations and monomial order.
For the low-crossing calculations, custom Python scripts used exact rational arithmetic. The workflow combined selected Wirtinger presentations, Fox matrices, ideal contractions, reduced Gröbner-basis calculations and direct membership checks.
Examples and family-wide calculations
Within the selected comparisons, the matching Alexander polynomials therefore did not force the same AG2 value. The examples support a narrower conclusion: the higher Alexander-ideal information captured by this construction can distinguish some representatives that the multivariable polynomial treats as equivalent up to Laurent units.
The paper also works out a specified torus-link family rather than stopping at individual examples. For every stated index satisfying 1 ≤ k ≤ ℓ, it determines the elementary ideals, their polynomial contractions and the reduced AGk Gröbner bases.
A two-component pretzel family receives a similarly explicit treatment. Its second contracted elementary ideal is reduced to a three-generated ideal in Q[x,y], with a Gröbner-basis construction built from cyclotomic pieces and interreduction. Before that interreduction, the uniform construction gives a Gröbner basis with at most four elements.
The paper also includes computations for three-component pretzel examples, but labels them exploratory computer evidence without proof. Those calculations are not used in the subsequent proofs.
A symmetry result with a clear boundary
A second major thread examines reciprocity, a symmetry relation for the algebraic objects involved. This analysis is carried out over Q for connected compact oriented 3-manifolds whose nonempty boundary is a disjoint union of tori.
Within that scope, the paper states determinant-divisor reciprocity for every k ≥ 0. It then shows that a Fitting ideal and its involutive image agree after localization at every height-one prime, giving agreement at the codimension-one level.
That codimension-one result does not by itself establish equality of the full ideals. To settle the remaining question for a particular calculation, the paper gives a finite Gröbner-basis test: reduce the reciprocal generators and check whether each has zero normal form.
The distinction is central to the paper’s claims. It does not assert full ideal reciprocity in general; any remaining discrepancy is confined to codimension at least two. The explicitly displayed computed ideals, however, are reported to be fully reciprocal.
What the calculations leave open
Taken together, the results provide a fixed Gröbner-basis representative, examples where AG2 separates links with matching multivariable Alexander polynomials, complete calculations for stated torus-link and pretzel families, and a finite test for full reciprocity in individual cases. The evidence remains tied to the selected representatives, algebraic conventions and specified families.
The low-crossing findings concern chosen two- and three-component representatives with at most 10 crossings, not all links. The three-component pretzel computations remain unproved, and the reciprocity analysis does not claim full ideal reciprocity in general.
The document is an arXiv version 1 preprint dated 26 August 2026.
Paper data and sources
Original title: Gröbner Bases for Alexander Fitting Ideals of Links
Authors: Takefumi Nosaka, Kotaro Shimizu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text