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On positivity of Roger--Yang skein algebras

Hiroaki Karuo

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 [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.

On positivity of Roger--Yang skein algebras

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 [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.
Paper Structure (13 sections, 11 theorems, 32 equations, 2 figures)

This paper contains 13 sections, 11 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Roger--Yang skein algebras and related works
  3. Positivity conjecture of skein algebras
  4. Main result and positivity conjecture
  5. Acknowledgments.
  6. Chebyshev polynomials
  7. Roger--Yang skein algebras and their classical versions
  8. $\mathcal{M}$-tangles and diagrams.
  9. Roger--Yang skein algebras.
  10. Bases of Roger--Yang skein algebras and positivity
  11. Lower bound of positive bases
  12. Partial evidence for positivity conjecture
  13. (Non-)positivity for small punctured surfaces

Key Result

Theorem 1

If a pair of normalized sequences $((Q_n(x)), (R_n(x)))$ is positive for a punctured surface of genus $g\geq 1$ with at least 3 punctures, then each of $Q_n(x)$ and $R_n(x)$ is a $\mathbb{Z}_{\geq0}[q^{\pm1/2}]$-linear combination of $\overline{T}_i(x)$ ($i=0,1,\dots, n$).

Figures (2)

  • Figure 1: In each local picture, the punctures are $p", p, p'$ from left to right. Left: $\beta\cdot p_\alpha\cdot \alpha$, Middle: $\overline{\alpha}\cdot p_\alpha$, Right: $\underline{\alpha}\cdot p_\alpha$
  • Figure 2: Picture of $(\mathbb{D},\mathcal{M}')$. The arc is $\alpha$ and the simple closed curve is $p_\alpha$.

Theorems & Definitions (26)

  • Theorem 1: Theorem \ref{['Thm_lower']}
  • Conjecture 2: Conjecture \ref{['Conj_posiRY']}
  • Example 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 3.1: BKL
  • Theorem 3.2: RY, MW21, MW, BKL
  • Theorem 3.3: MW, see also FST
  • Proposition 4.1
  • proof
  • ...and 16 more