Skein relations on p unctured surfaces
This thesis studies skein relations in c luster algebras arising from punctured surfaces. We introduce identit ies expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corresponding to compatibl e arcs. Incompatibility arises from phenomena such as intersections, self-intersections, and opposite taggings at punctures. To establish these identities, we develop a combinatorial framework that relates l oop 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 c onditions. This extends existing work on surface cluster algebras by incorporating punctures in the interior of the surface, thereby enlar ging the class of cluster algebras for which such skein relations and bases can be constructed.
More information on the thesis
DESCRIPTION: Skein relations on punctured surfaces This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corre sponding to compatible arcs. Incompatibility arises from phenomena su ch as intersections, self-intersections, and opposite taggings at pun ctures. To establish these identities, we develop a combinatorial fra mework that relates loop graphs to certain representations. These ske in 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 positivit y and compatibility conditions. This extends existing work on surface cluster algebras by incorporating punctures in the interior of the s urface, thereby enlarging the class of cluster algebras for which suc h skein relations and bases can be constructed.More information on th e thesis END:VEVENT END:VCALENDAR