2024/09/08 by Esther Banaian, Banaian, Esther, Wonwoo Kang +3 · 2 citations
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2409.04957
We investigate skein relations in cluster algebras from punctured surfaces, extending the work of Çanakçi-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by Oğuz-Yıldırım and Pilaud-Reading-Schroll, we provide explicit formulas for these relations. This work demonstrates that the punctured analogues of the bangle and bracelet functions form spanning sets for cluster algebras associated with a punctured surfaces. For surfaces with boundary and closed surfaces of genus 0, we further show that the bangles and bracelets form bases.