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Lattice paths on snake graphs for q-deformed rationals

  • Jérôme Germoni
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Abstract

We give an elementary, straightforward proof of Ovenhouse's interpretation of the denominator and the numerator of a (right) q-deformed rational in terms of weighted lattice paths on a "snake graph", and we give an analogous result for the left q-deformed rational introduced by Bapat, Becker, and Licata. The key ingredient is a simple formula for the q-deformation of the matrix of the composition of two homographies of the form z → a + 1/z.

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Cite

BibTeXRIS
BibTeX
@misc{germoni2026lattice,
  title = {{Lattice paths on snake graphs for q-deformed rationals}},
  author = {Jérôme Germoni},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09953},
  eprint = {2610.09953},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05781778v1},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Lattice paths on snake graphs for q-deformed rationals
AU  - Jérôme Germoni
PY  - 2026
DA  - 2026/10/06
PB  - HAL
DO  - 10.48550/arxiv.2610.09953
UR  - https://hal.science/hal-05781778v1
AB  - We give an elementary, straightforward proof of Ovenhouse's interpretation of the denominator and the numerator of a (right) q-deformed rational in terms of weighted lattice paths on a "snake graph", and we give an analogous result for the left q-deformed rational introduced by Bapat, Becker, and Licata. The key ingredient is a simple formula for the q-deformation of the matrix of the composition of two homographies of the form z → a + 1/z.
ER  -