HEXAGONbeta

Counterexamples to Source–Sink Recursions for Denominator Vectors

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.RA — Rings and Algebras

Contributed by Qiyue Tang ↗

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

Reading and Stella’s source–sink recursions for denominator vectors, Conjectures 2.7 and 2.8, fail for an infinite family of four-vertex exchange matrices. The member realized by an unpunctured annulus with one marked point on one boundary and three on the other also refutes their Theorem 2.11.

Provenance statement

The research run using GPT-6-Pro lasted approximately two hours in total and included the work that produced the counterexamples in this paper. Codex was used to check the arguments and revise the exposition. A review using Rethlas-OSV is planned for a later revision.

References

  1. C. Ceballos and V. Pilaud, Denominator vectors and compatibility degrees in cluster algebras of finite type , Trans. Amer. Math. Soc. 367 (2015), 1421--1439. arXiv:1302.1052 (https://arxiv.org/abs/1302.1052).arXiv
  2. S. Fomin, M. Shapiro, and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes , Acta Math. 201 (2008), 83--146. doi:10.1007/s11511-008-0030-7 (https://doi.org/10.1007/s11511-008-0030-7).DOI
  3. S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations , J. Amer. Math. Soc. 15 (2002), 497--529. arXiv:math/0104151 (https://arxiv.org/abs/math/0104151).arXiv
  4. S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification , Invent. Math. 154 (2003), 63--121. arXiv:math/0208229 (https://arxiv.org/abs/math/0208229).arXiv
  5. S. Fomin and A. Zelevinsky, Cluster algebras IV: Coefficients , Compos. Math. 143 (2007), 112--164. arXiv:math/0602259 (https://arxiv.org/abs/math/0602259).arXiv
  6. C. Fu and B. Keller, On cluster algebras with coefficients and 2-Calabi--Yau categories , Trans. Amer. Math. Soc. 362 (2010), 859--895. arXiv:0710.3152 (https://arxiv.org/abs/0710.3152).arXiv
  7. N. Reading and S. Stella, Initial-seed recursions and dualities for dd-vectors , Pacific J. Math. 293 (2018), 179--206. doi:10.2140/pjm.2018.293.179 (https://doi.org/10.2140/pjm.2018.293.179).DOI
  8. D. Rupel and S. Stella, Some consequences of categorification , SIGMA 16 (2020), 007 (https://doi.org/10.3842/SIGMA.2020.007), 8 pp.DOI
  9. T. Yurikusa, Denominator vectors and dimension vectors from triangulated surfaces , J. Algebra 641 (2024), 620--647. arXiv:2305.08097v2 (https://arxiv.org/abs/2305.08097v2).arXiv

Version history

  1. v1 · 5 pagesSubmitted by Qiyue TangInitial depositCurrentNov 07, 2026