Research Paper#Cluster Algebras, Punctured Surfaces, Skein Relations🔬 ResearchAnalyzed: Jan 3, 2026 17:12
Skein Relations in Punctured Surface Cluster Algebras
Published:Dec 30, 2025 20:01
•1 min read
•ArXiv
Analysis
This paper extends the study of cluster algebras, specifically focusing on those arising from punctured surfaces. It introduces new skein-type identities that relate cluster variables associated with incompatible curves to those associated with compatible arcs. This is significant because it provides a combinatorial-algebraic framework for understanding the structure of these algebras and allows for the construction of bases with desirable properties like positivity and compatibility. The inclusion of punctures in the interior of the surface broadens the scope of existing research.
Key Takeaways
- •Introduces skein-type identities for cluster algebras on punctured surfaces.
- •Develops a combinatorial-algebraic framework relating loop graphs to representations.
- •Enables the construction of bases with positivity and compatibility conditions.
- •Extends existing work by incorporating punctures in the interior of the surface.
Reference
“The paper introduces skein-type identities expressing cluster variables associated with incompatible curves on a surface in terms of cluster variables corresponding to compatible arcs.”