Relationships between skein algebras

We will examine the multiplicative structure of two skein algebras—the usual Kauffman bracket skein algebra of a surface (generated by loops) and a generalization of it due to Roger-Yang (generated by loops and arcs).  In joint work with Chloe Marple, we found an unexpected homomorphism between the usual skein algebra for a closed torus and the Roger-Yang skein algebra for a twice-punctured annulus.  In this talk, I’ll discuss how we  used the homomorphism to help compute representations and structural constants of the Roger-Yang skein algebra for a twice-punctured annulus, and  whether there might be similar relationships between skein algebras for other surfaces.