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arXiv · 2409.04957

Skein relations for punctured surfaces

Esther Banaian·Wonwoo Kang·Elizabeth Kelley
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Abstract

We investigate skein relations in cluster algebras from punctured surfaces, extending the work of \c{C}anak\c{c}i-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by O{\u{g}}uz-Y{\i}ld{\i}r{\i}m and Pilaud-Reading-Schroll, we provide explicit formulas for these relations. This work demonstrates that the punctured analogues of the bangle and bracelet functions form spanning sets for cluster algebras associated with a punctured surfaces. For surfaces with boundary and closed surfaces of genus 0, we further show that the bangles and bracelets form bases.

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Esther Banaian, Wonwoo Kang, Elizabeth Kelley. 2024-09-08. Skein relations for punctured surfaces. https://arxiv.org/abs/2409.04957

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