Skein relations on punctured surfaces

TL;DR

This thesis develops skein relations for cluster algebras from punctured surfaces, providing new identities that relate incompatible and compatible cluster variables, and establishing bases with positivity properties.

Contribution

Introduces skein-type identities for punctured surface cluster algebras and a combinatorial-algebraic framework relating loop graphs to representations, extending previous work.

Findings

Skein relations express incompatible cluster variables in terms of compatible ones.

Framework relates loop graphs to representations for establishing identities.

Proves existence of positive, compatible bases for punctured surface cluster algebras.

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.