PT - JOURNAL ARTICLE
AU - Thurston, Dylan Paul
TI - Positive basis for surface skein algebras
AID - 10.1073/pnas.1313070111
DP - 2014 Jul 08
TA - Proceedings of the National Academy of Sciences
PG - 9725--9732
VI - 111
IP - 27
4099 - http://www.pnas.org/content/111/27/9725.short
4100 - http://www.pnas.org/content/111/27/9725.full
SO - Proc Natl Acad Sci USA2014 Jul 08; 111
AB - 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.We show that the twisted SL2 skein algebra of a surface has a natural basis (the bracelets basis) that is positive, in the sense that the structure constants for multiplication are positive integers.