11 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…

Arm-property conjecture. The identity

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…

Let $n\geq 2$, let $\widetilde{G}_{n}^{Mb_{n,1}}$ denote the remaining block in the decomposition of the Gram matrix of type $(Mb)_1$, and let $T_{2k}$ be the Chebyshev polynomial…

Let $n\geq 1$, let $D_n^{(Mb)_1}$ denote the Gram determinant of type $(Mb)_1$, let $T_k$ be the $k$th Chebyshev polynomial of the first kind, and set $d=-A^2-A^{-2}$. Ishii–Morton…

Gram locus conjecture. The Gram locus $\mathcal{G}(\mathcal{B})$ is given by

Let $G_n$ be an easy quantum group and $G_n^+$ its liberation, with inclusion $G_n\subset G_n^+$. This inclusion gives a planar algebra module in the sense of Jones, and the Gram m…

Let $G_n$ be an easy quantum group, let $4\mathcal P(k)4$ denote the set of partitions of $k$ points, and let $G_{kn}$ be the associated Gram matrix. A decomposition of the determi…

Barad's factorization conjecture. The determinant satisfies

In the restricted two-boundary Temperley–Lieb algebra, let ${\cal G}^{\rm uu}_{N,0}$ be the Gram matrix in the sector with $N$ sites and no through-lines, and let $n_{\rm b}$ be th…