Skein relations on punc tured surfaces
This thesis studies skein relations in cluster algebras arising from punctured surfaces. We introduce identities exp ressing cluster variables associated with incompatible curves on a su rface in terms of cluster variables corresponding to compatible arcs. Incompatibility arises from phenomena such as intersections, self-in tersections, and opposite taggings at punctures. To establish these i dentities, we develop a combinatorial framework that relates loop gra phs to certain representations. These skein relations can then be app lied to investigate structural properties of cluster algebras from pu nctured surfaces. In particular, they can be used to prove the existe nce of bases satisfying natural positivity and compatibility conditio ns. This extends existing work on surface cluster algebras by incorpo rating punctures in the interior of the surface, thereby enlarging th e class of cluster algebras for which such skein relations and bases can be constructed.
More information on the thesis
DESCRIPTION: This thesis studies skein relations in cluster algebras a rising from punctured surfaces. We introduce identities expressing cl uster variables associated with incompatible curves on a surface in t erms of cluster variables corresponding to compatible arcs. Incompati bility arises from phenomena such as intersections, self-intersection s, and opposite taggings at punctures. To establish these identities, we develop a combinatorial framework that relates loop graphs to cer tain representations. These skein relations can then be applied to in vestigate structural properties of cluster algebras from punctured su rfaces. In particular, they can be used to prove the existence of bas es satisfying natural positivity and compatibility conditions. This e xtends existing work on surface cluster algebras by incorporating pun ctures in the interior of the surface, thereby enlarging the class of cluster algebras for which such skein relations and bases can be con structed. More information on the thesis Skein relations on p unctured surfaces END:VEVENT END:VCALENDAR