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.
A Generalization of the Gram determinant of type A
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abstract
The Gram determinant of type $A$ was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type $A$ by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type $Mb$ evaluated in the Klein bottle.
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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.