On positivity of Roger--Yang skein algebras

TL;DR

This paper extends the positivity conjecture from Kauffman bracket skein algebras to Roger--Yang skein algebras, proposing positive bases using Chebyshev polynomials and exploring their relation to algebra centers at roots of unity.

Contribution

It introduces explicit positive bases for Roger--Yang skein algebras using Chebyshev polynomials and connects these bases to the algebra centers at roots of unity.

Findings

Chebyshev polynomials form positive bases for Roger--Yang skein algebras.

The polynomials provide a lower bound as per previous works.

A relation between these polynomials and algebra centers at roots of unity is established.

Abstract

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 [L\^e18] 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.