Skein relations for punctured surfaces
Abstract
We investigate skein relations in cluster algebras from punctured surfaces, extending the work of Canakci-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by Oguz-Yldrm 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.
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