Traces on the skein algebra of the torus

For a surface F, the Kauffman bracket skein module of F × [ 0 , 1 ] , denoted K ( F ) , admits a natural multiplication which makes it an algebra. When specialized at a complex number t, nonzero and not a root of unity, we have K t ( F ) , a vector space over C . In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space K t ( T 2 ) has five distinct traces. One trace, the Yang–Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on K t ( T 2 ) correspond to the four singular points of the moduli space of flat SU ( 2 ) -connections on the torus.