Traces on the skein algebra of the torus

摘要

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 numbert, 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.