Chebyshev polynomials and Gram determinants from the Möbius band
Anthony Christiana, Dionne Ibarra, Gabriel Montoya-Vega
TL;DR
The paper investigates how Chebyshev polynomials encode Gram determinants arising from crossingless connections on a Möbius band in knot theory. It proves a product relation $S_{M_k}(d)=\prod_{i=0}^{k-1} T_{2^i}(d)$ for $M_k=2^k-1$ and develops a framework linking these polynomials to Gram determinants of type $(Mb)_1$, including a concrete expression for the related determinant $\det(\tilde{G}_n^{Mb_{n,1}})$ and a provable factor $S_1(d)^{2k}$ with $k=\binom{2n}{n-2}$. The authors provide a computational approach to evaluate the Gram matrix symbolically and restate Qi Chen's conjectured closed form for $D_n^{(Mb)_1}$, offering a concrete path toward a full proof. They also outline future directions, notably a potential proof strategy using Jones-Wenzl idempotents and Chebyshev identities to establish the full determinant formula. $S_{M_k}(d)=\prod_{i=0}^{k-1} T_{2^i}(d)$ and the determinant factors are central to the connection between knot invariants and Chebyshev theory in this work.
Abstract
This article explores the connection between Chebyshev polynomials and knot theory, specifically in relation to Gram determinants. We reveal intriguing formulae involving the Chebyshev polynomial of the first and second kind. In particular we show that for Mersenne numbers, $M_k=2^k-1$ where $k\geq 2$, the $M_k$-th Chebyshev polynomial of the second kind is the product of Chebyshev polynomials of the first kind. We then discuss the Gram determinant of type $(Mb)_1$, restate the conjecture of its closed formula in terms of mostly products of Chebyshev polynomials of the second kind, and prove a factor of the determinant that supports the conjecture. We also showcase an algorithm for calculating the Gram determinant's corresponding matrix. Furthermore, we restate Qi Chen's conjectured closed formula for the Gram determinant of type Mb and discuss future directions.