Lower and Upper Bounds for Positive Bases of Skein Algebras

TL;DR

This paper investigates the constraints on positive bases of skein algebras, demonstrating they are bounded by Chebyshev polynomials, and uniquely identifying the Chebyshev sequence as the sole positive basis for the closed torus.

Contribution

It establishes bounds for positive bases in skein algebras and characterizes the unique positive basis for the closed torus using Chebyshev polynomials.

Findings

Positive bases are bounded by Chebyshev polynomials.

The Chebyshev polynomials of type one uniquely form a positive basis for the closed torus.

The results provide bounds and uniqueness criteria for positive bases in skein algebras.

Abstract

We show that the if a sequence of normalized polynomials gives rise to a positive basis of the skein algebra of a surface, then it is sandwiched between the two types of Chebyshev polynomials. For the closed torus, we show that the normalized sequence of Chebyshev polynomials of type one $(\hat{T}_n)$ is the only one which gives a positive basis.

Equations

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