39 problems

Web compatibility conjecture. Two cluster or coefficient variables are compatible if and only if their product is a web invariant.

Arborization conjecture. The forest form of a cluster monomial can be found from the web form via the arborization algorithm.

Alternating-sign polygon reduction conjecture. Any $2n$-gon of sign type $(+,-,+,\ldots,-)$ can be expanded as a sum of webs consisting of polygons of smaller sizes and $2n$-gons o…

In the setting of a punctured surface $\Sigma$, let $\mathcal{X}$-webs denote the webs introduced by Shen–Sun–Weng, and let…

Let $\mathcal{W}$ be the set of rank $2$ webs $W(R,S,T,V)$ on the Grassmannian $\operatorname{Gr}(k,n)$, where $R,S,T$ are cyclically separated subsets and $V$ is disjoint from the…

Positivity characterization conjecture. Traces of all $2$-webs in $H$ are positive if and only if, up to gauge equivalence, one of the following conditions holds:

Let $\hat{S}$ be a compact oriented surface with marked boundary points and punctures. Let $\mathcal{P}_{{\rm PGL}_3,\hat{S}}^+(\mathbb{Z}^t)$ denote the integer tropical points sa…

Let $\tilde{G}$ and $\tilde{G}'$ be tangled tree diagrams related by crossing changes, and let $G\in\mathsf{Tree}_{\mathfrak{sp}_4,\Sigma}$ be a tree-type elementary web. Crossing-…

Let $\Sigma$ be an unpunctured marked surface. Let $\mathsf{EWeb}_{\mathfrak{sp}_4,\Sigma}$ be the set of elementary webs, let…

The paper studies ${\rm SL}_3$ web immanants and their relation to the degree-three part of the Grassmannian coordinate ring. Degree-two web immanant conjecture. The analogue of Th…

Let $\mathbbm{k}$ be a field, and let \text{\boldmatha} \in \mathbbm{Z}^{r} and \text{\boldmathb} \in \mathbbm{Z}^{s} be the parameters indexing the relevant objects, with…

General web-series conjecture. For every such web $W_i$, there exists a natural way to associate a web series to it.

Almost-arborizable web-series conjecture. For every almost arborizable $W$, the quantities $A(W)$ and $B(W)$ agree with the corresponding $A$ and $B$ in the cluster series associat…

Web-series existence conjecture. A web series exists for every ray.

Fomin–Pylyavskyy conjecture. If a web is arborizable, then its invariant $[W]$ is a cluster variable.

Let the type $C$ web category be presented by the relations in … can be removed, apart from such trivial special cases. The conjecture asserts that this displayed presentation is m…

Rank-inequality conjecture. One has

Rank-separation conjecture. If $c_{\widetilde{w}}^{w'}\neq 0$, then

Tableau-order support conjecture. If $c_{\widetilde{w}}^{w'}\neq 0$, then $\widetilde{w}\prec_T w'$.

Specht–web unitriangularity conjecture. The transition matrix between the Specht and $\mathfrak{sl}_3$ web bases is unitriangular with respect to any total order completing the tab…

An annular web is a web in the annulus, regarded as an object of the annular Khovanov–Rozansky category. An essential unknot is a circle winding once around the annulus, and it may…

Annular web decomposition conjecture. Any annular $\mathfrak{sl}_\infty$ web is isomorphic to a direct sum of labeled, concentric circles.

Let $K$ be a web embedded in $\mathbb{R}^{3}$, and let $\Jsharp(K)$ be its instanton homology over the field $\mathbb{F}=\mathbb{Z}/2$. A Tait coloring of $K$ is a Tait coloring of…

Assume there is a bijection between generalized oscillating tableaux of length $k$ with $n$ parts, whose final component is $(m,\ldots,m)$ for some $m\in\mathbb{Z}$, and…

Let a generalized oscillating tableau have length $k$ and $n$ parts, with final component $(m,\ldots,m)$ for some $m\in\mathbb{Z}$. Consider $\operatorname{sl}(n)$ webs with $k$ bo…