3 problems

Let $R=\mathbb{Z}[A^{\pm1},w,x,y,z]$. For $n\geq1$, let $D_n^{\mathit{Mb}}(d,w,x,y,z)$ be the Gram determinant of type $\mathit{Mb}$. Let $T_k$ and $S_k$ denote the Chebyshev polyn…

Let $R=\mathbb{Z}[A^{\pm1},w,x,y,z]$. For $n\geq 1$, let $D_n^{(Mb)_1}$ denote the Gram determinant of type $(Mb)_1$, where the Gram matrix is formed from crossingless-connection d…

Let $Mb_{n,1}$ be the set of crossingless-connection diagrams between $2n$ marked boundary points of the Möbius band, with exactly one curve intersecting the crosscap. Let…