14 problems

Suppose that a planar diagram of an oriented link $L$ intersects the vertical line $x=0$ at $(0,-b)$, $(0,-a)$, $(0,a)$, and $(0,b)$, where $0<a<b$. Suppose that $L$ is directed to…

Let $G$ and $G'$ be connected simple plabic graphs whose quivers $Q_G$ and $Q_{G'}$ are mutation equivalent. Let $L^{\operatorname{plab}}_G$ and $L^{\operatorname{plab}}_{G'}$ be t…

Let $G$ be a plabic graph with $\operatorname{c}(G)$ connected components and $n$ interior faces. Let $L^{\operatorname{plab}}_G$ be its associated plabic graph link, and let…

Let $G$ be a simple plabic graph with $\operatorname{c}(G)$ connected components. Let $Q_G$ be its planar dual quiver, let $L^{\operatorname{plab}}_G$ be the associated plabic grap…

The $Q^r$ shift conjecture. For all odd integers $a,b,c$ and symmetric representations $[r]$,

The normalized $Q^1$ periodicity conjecture. For the indicated values of $c$ and $r$,

The $9_{46}$ transformed-factor conjecture.

The $9_{46}$ odd-index divisibility conjecture. For odd positive integers $i>1$,

The $9_{35}$ divisibility conjecture. For positive integers $i>1$,

Eigenvalue conjecture. The quantum $\mathcal{R}_i$-matrices are fully determined by the eigenvalues of the universal $\mathcal{R}$ matrix. Equivalently, to establish invariance und…

Let $Z_{[33]}^{(p)}$ be the quadratic differentials associated with the rectangular representation $[33]=[3^2]$, and let $[n]$ denote the quantum number. Figure-eight [33] expansio…

Universal difference conjecture. The difference has the form

Quantization conjecture for super-A-polynomials. The $Q$-deformed $A$-polynomial is identified with the augmentation polynomial of knot contact homology. This proposed identificati…

Let $\mathcal{K}$ be a knot, let $H_R^{\mathcal{K}}(A\mid q)$ denote its HOMFLY polynomial in representation $R$, and let $|R|$ be the size of $R$. The knot complement is…