We investigate skein relations in cluster algebras from punctured surfaces, extending the work of \c{ C}anak\c{ c}i-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by O{ \u{ g}}uz-Y{ \i}ld{ \i}r{ \i}m 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.
Categories: math.CO