The paper proves that the (2^k-1)th second-kind Chebyshev polynomial factors as a product of first-kind Chebyshev polynomials, restates two Gram determinant conjectures, and proves a supporting divisibility factor that currently has a proof gap.
On a new Gram determinant from the M\"obius band
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abstract
Gram determinants earned traction among knot theorists after E. Witten's presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by N. Reshetikhin and V. Turaev, several mathematicians have explored this line of research ever since. Gram determinants came into play by W. B. Raymond Lickorish's skein theoretic approach to the invariant. The construction of different bilinear forms is possible through changes in the ambient surface of the Kauffman bracket skein module. Hence, different types of Gram determinants have arisen in knot theory throughout the years; some of these determinants are discussed here. In this article, we introduce a new version of such a determinant from the M\"obius band and prove some important results about its structure. In particular, we explore its connection to the annulus case and factors of its closed formula.
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Chebyshev polynomials and Gram determinants from the M\"obius band
The paper proves that the (2^k-1)th second-kind Chebyshev polynomial factors as a product of first-kind Chebyshev polynomials, restates two Gram determinant conjectures, and proves a supporting divisibility factor that currently has a proof gap.