Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened T-Shirt

In 1999, Charles Frohman and Răzvan Gelca discovered a much celebrated product-to-sum formula for the multiplicative operation in this algebra. In 2018, the first two authors along with Sujoy Mukherjee, Marithania Silvero, and Xiao Wang studied the multiplicative structure of the Kauffman bracket skein algebra of the four-holed sphere. They presented an algorithm to compute the product of any two elements of the algebra, and gave an explicit formula for some families of curves. Further, they conjectured the existence of a positive basis for the algebra. This is now called strong positivity of the basis of the Kauffman bracket skein algebra. In this lecture we discuss their work in detail and briefly touch upon Pierrick Bousseau’s work in resolving this conjecture.