Positive basis for surface skein algebras

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Abstract
Significance: The Jones polynomial of knots is one of the simplest and most powerful knot invariants, at the center of many recent advances in topology; it is a polynomial in a parameter q . The skein algebra of a surface is a natural generalization of the Jones polynomial to knots that live in a thickened surface. In this paper, we propose a basis for the skein algebra. This basis has positivity properties when q is set to 1, and conjecturally for general values of q as well. This is part of a more general conjecture for cluster algebras, and suggests the existence of well-behaved higher-dimensional structures.

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