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PreprintCC BY 4.0arXiv

Gröbner Bases for Alexander Fitting Ideals of Links

  • Takefumi Nosaka
  • Kotaro Shimizu
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Abstract

Multivariable Alexander theory produces Fitting ideals without preferred generators. After fixing a coefficient field and a monomial order, we represent these ideals by reduced Gröbner bases of their polynomial contractions, obtaining Alexander--Gröbner invariants of links. Over ℚ, for the maximal free-abelian coefficient system of a connected compact oriented 3-manifold whose nonempty boundary is a disjoint union of tori, we deduce determinant-divisor reciprocity from Blanchfield duality. We compute these invariants for torus links and low-crossing links, and we determine AG2 for a two-component pretzel family.

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Cite

BibTeXRIS
BibTeX
@misc{nosaka2026gröbner,
  title = {{Gröbner Bases for Alexander Fitting Ideals of Links}},
  author = {Takefumi Nosaka and Kotaro Shimizu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.25725},
  eprint = {2608.25725},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25725},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Gröbner Bases for Alexander Fitting Ideals of Links
AU  - Takefumi Nosaka
AU  - Kotaro Shimizu
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2608.25725
UR  - https://arxiv.org/abs/2608.25725
AB  - Multivariable Alexander theory produces Fitting ideals without preferred generators. After fixing a coefficient field and a monomial order, we represent these ideals by reduced Gröbner bases of their polynomial contractions, obtaining Alexander--Gröbner invariants of links. Over ℚ, for the maximal free-abelian coefficient system of a connected compact oriented 3-manifold whose nonempty boundary is a disjoint union of tori, we deduce determinant-divisor reciprocity from Blanchfield duality. We compute these invariants for torus links and low-crossing links, and we determine AG_2 for a two-component pretzel family.
ER  -