On positivity of Roger--Yang skein algebras

We generalize the positivity conjecture on (Kauffman bracket) skein algebras to Roger–Yang skein algebras. To generalize it, we use explicit polynomials like Chebyshev polynomials of the first kind to give candidates of positive bases. Moreover, the polynomials form a lower bound in the sense of [Lê18] and [LTY21]. We also discuss a relation between the polynomials and the centers of Roger–Yang skein algebras when the quantum parameter is a complex root of unity.