Traces on the skein algebra of the torus

Michael McLendon

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Traces on the skein algebra of the torus

Michael McLendon

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Abstract

For a surface $F$, the Kauffman bracket skein module of $F \times [0,1]$, 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 $\mathbb{C}$. 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 each of the four $\mathbb{Z}_2$ homology classes of the torus.

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Michael McLendon. 2006-03-14. Traces on the skein algebra of the torus. https://arxiv.org/abs/math/0603343

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