Traces on the skein algebra of the torus

2006/03/14 by Michael McLendon, McLendon, Michael
Mathematics · #57M27 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M27

paper · pdf · doi:10.48550/arxiv.math/0603343

8 pages, 1 figure

arxiv created 2006/03/14 · arxiv updated 2009/12/01

Abstract

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(T²) 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(T²) correspond to each of the four ℤ₂ homology classes of the torus.

Related

• The Yang-Mills Measure in the Kauffman Bracket Skein Module
• Dimension and Trace of the Kauffman Bracket Skein Algebra
• The Kauffman skein algebra of the torus
• A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces
• Multiplicative structure of Kauffman bracket skein module quantizations