TY - JOUR
T1 - Positive basis for surface skein algebras
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 9725
LP - 9732
DO - 10.1073/pnas.1313070111
VL - 111
IS - 27
AU - Thurston, Dylan Paul
Y1 - 2014/07/08
UR - http://www.pnas.org/content/111/27/9725.abstract
N2 - 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.
ER -