Urbana Campus Research Calendar (OVCRI)

We investigate skein relations in cluster algebras arising from punctured surfaces, building on the work of Çanakcı-Schiffler and Musiker-Williams for unpunctured surfaces. Using the combinatorial expansion formulas by Oğuz-Yıldırım and Pilaud-Reading-Schroll, we derive explicit expressions for these relations. Our approach provides a detailed account of the y-monomials involved and shows that one term consistently has a coefficient of 1. This work demonstrates that the punctured analogues of the bangle and bracelet bases form a spanning set for the cluster algebra associated with punctured surfaces.