Rotundus: triangulations, Chebyshev polynomials, and Pfaffians

TL;DR

This paper introduces a cyclically invariant polynomial related to the continuant, showing it can be computed as a Pfaffian, and explores its connections to Chebyshev polynomials and Diophantine equations.

Contribution

It presents a novel polynomial invariant, establishes its Pfaffian representation, and extends classical results to a cyclically invariant setting with applications to Chebyshev polynomials.

Findings

The polynomial can be computed as a Pfaffian of a skew-symmetric matrix.

An analog of Conway and Coxeter's Diophantine result is proved.

Chebyshev polynomials of the first kind are shown to arise as Pfaffians.

Abstract

We introduce and study a cyclically invariant polynomial which is an analog of the classical tridiagonal determinant usually called the continuant. We prove that this polynomial can be calculated as the Pfaffian of a skew-symmetric matrix. We consider the corresponding Diophantine equation and prove an analog of a famous result due to Conway and Coxeter. We also observe that Chebyshev polynomials of the first kind arise as Pfaffians.