Skein relations on punctured surfaces

Abstract

This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce skein-type identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corresponding to compatible arcs. Incompatibility arises from phenomena such as intersections, self-intersections, and opposite taggings at punctures. To establish these identities, we develop a combinatorial-algebraic framework that relates loop graphs to certain representations. These skein relations can then be applied to investigate structural properties of cluster algebras from punctured surfaces. In particular, they can be used to prove the existence of bases satisfying natural positivity and compatibility conditions. This extends existing work on surface cluster algebras by incorporating punctures in the interior of the surface, thereby enlarging the class of cluster algebras for which such skein relations and bases can be constructed.