SIGMA 22 (2026), 019, 42 pages      arXiv:2312.06148      https://doi.org/10.3842/SIGMA.2026.019

Matrix Formulae and Skein Relations for Quasi-Cluster Algebras

Cody Gilbert ^a, McCleary Philbin ^b and Kayla Wright ^c
a) Department of Mathematics, Saint Louis University, Saint Louis, MO, USA
^b) Department of Mathematics, University of Wisconsin - River Falls, River Falls, WI, USA
^c) Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

Received May 14, 2025, in final form January 30, 2026; Published online February 27, 2026

Abstract
In this paper, we give matrix formulae for non-orientable surfaces that provide the Laurent expansion for quasi-cluster variables, generalizing the orientable surface matrix formulae by Musiker-Williams. We additionally use our matrix formulas to prove the skein relations for the elements in the quasi-cluster algebra associated to curves on the non-orientable surface.

Key words: cluster algebra; snake graphs; triangulated surfaces.

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