14 problems

2BTL representation-identification conjecture. For $n>0$, the irreducible representation $W^{(N,n)}_{\epsilon_1,\epsilon_2}$ is equivalent to $V^{(N,n)}_{\epsilon_1,\epsilon_2}$, a…

Let $\Lambda_n$ index the cell modules $\Delta(\lambda)$ and simple modules $L(\mu)$ of the orientifold quiver Temperley--Lieb algebra, and let $\langle k\rangle$ denote grading sh…

Convergence conjecture. This rewriting system is convergent, meaning that it is terminating and confluent.

Let $\kappa$ be the coefficient field, let $\operatorname{Ch}^-(\mathsf{BN}_n)$ be the homotopy category of bounded-above complexes of $n$-strand Bar–Natan 1-morphisms, and let…

Let $w_{0},w_{1}$ be link states, let $\mathcal{S}(w_{0})$ and $\mathcal{S}(w_{1})$ be the associated spin-chain representations, and define … by…

Temperley–Lieb–Virasoro braided equivalence conjecture. There is an equivalence of braided tensor categories

Affine Temperley–Lieb quotient conjecture. There is an isomorphism of algebras

At $q=1$, write the Type BII sum as … with $S_{N,j}\in\mathbb{N}$. Type BII first-coefficient conjecture. … The paper reports verification through $N=20$, but no general proof or r…

Let $si$ be the ground-state vector, and let $S^{\mathrm{X}}_{N}(q,Q)$ denote the sum of all its components on the Kazhdan–Lusztig basis of boundary type $\mathrm{X}$, where…

For $n\geq 3$, let $\mathcal{B}_n^{\pm}$ be the specialized algebra, let $\mathcal{I}_n^{\pm}$ be its defining ideal, and let…

Let ${\rm PTL}_n$ be the Partition Temperley–Lieb algebra, let $\mathsf{P}_n$ denote the set-partition monoid on $\{1,\ldots,n\}$, and let $F$ be a reduced word in the generators u…

Gram-matrix parity conjecture. One has

Barad's factorization conjecture. The determinant satisfies

Let $A_{\textsc{HT}}(N;\pi)$ be the number of half-turn-invariant FPLs of size $N$ with half-turn-invariant link pattern $\pi$, and let $A_{\textsc{HT}}(N)$ be the total number of…