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arXiv · 2606.02164

Band bases as common triangular bases in cluster algebras from surfaces

Fan Qin·Chao Shen
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Abstract

We consider the skein algebra of an unpunctured marked surface. Thurston previously constructed its band basis topologically. We show that this band basis coincides with the common triangular basis, which is a Kazhdan-Lusztig type basis for quantum cluster algebras analogous to the dual canonical basis of quantum groups. Our result confirms a conjecture of Thurston. It also provides new cases for the existence of the common triangular basis. In addition, we discover a phenomenon where certain unknots are arranged in configurations resembling beads on a necklace.

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Fan Qin, Chao Shen. 2026-06-01. Band bases as common triangular bases in cluster algebras from surfaces. https://arxiv.org/abs/2606.02164

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