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PreprintCC BY 4.0arXiv

An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties

  • Joe Boninger
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Abstract

We define an invariant of alternating links---a homogeneous, four-variable Laurent polynomial---that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has M-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.

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Cite

BibTeXRIS
BibTeX
@misc{boninger2026alexander,
  title = {{An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties}},
  author = {Joe Boninger},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28484},
  eprint = {2608.28484},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28484},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties
AU  - Joe Boninger
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28484
UR  - https://arxiv.org/abs/2608.28484
AB  - We define an invariant of alternating links---a homogeneous, four-variable Laurent polynomial---that encodes the symmetrized Alexander polynomial, the signature, and other topological data. Along the way, we extend a spanning tree formulation of the Alexander polynomial due to Murasugi and Stoimenow from special alternating links to all alternating links. This project is motivated by Fox's trapezoidal conjecture; accordingly, we prove certain sequences associated to our invariant are trapezoidal for all alternating links. We also conjecture our polynomial has M-convex support, and that it satisfies symmetry and log-concavity properties. We prove a partial symmetry result.
ER  -