34 problems

Let the skein algebra of a marked surface have frozen variables, and consider its localization at those frozen variables. The skein algebra carries a (quantized) band basis and a c…

Localized skein–cluster conjecture. Under these assumptions,

Let $\mathfrak{S}$ be a triangulable pb surface, let $\mathcal V$ be the vertex set of the associated quantum seed, and let $\mathcal V_{\mathrm{mut}}$ be its mutable vertices. Wri…

Let $\Gamma=H_1(\widetilde{\Sigma};\mathbb{Z})$ have its standard intersection pairing, and let … where $\sigma$ is the $\mathbb{Z}/2$-deck transformation action. Let $Q_{2\Gamma}$…

Let $\Sigma=\Sigma_{g,p}$, let $Sk^{ta}(\Sigma)$ be the tagged skein algebra, let $U(\Sigma)$ be the upper cluster algebra, and let $\nu:Sk^{ta}(\Sigma)\to U(\Sigma)$ be the natura…

Let $H_g$ be the genus $g$ handlebody, let $\rho\in \mathfrak{X}_{{\rm SL}_3(\mathbb{C})}(H_g)$ be represented by an irreducible representation, and let…

Let $\mathfrak{S} = \Sigma \setminus \mathcal{P}$ be a triangulable marked surface, let $\Delta$ be an ideal triangulation, and let $\hbar \in \mathbb{R}_{>0}$. For an essential si…

Let $S=S_{g,n}$ be a surface of genus $g\leq 1$ with $n>0$ boundary components, and let $(\widetilde{G},N)$ be the group and representation associated to a pants decomposition of…

Let $T^2-D$ be the once-punctured torus, and let $\widehat{\operatorname{Sk}}(T^2-D)$ be its completed skein algebra. For $\mathbf{x},\mathbf{y}\in\mathbb{Z}^2$, let…

Bonahon–Wong–Yang volume conjecture. In the above setup,

Let $S$ be an oriented surface of finite topological type. For each puncture, let $[P_i]$ denote the skein represented by a small loop around that puncture with weight…

Let $\Sigma$ be a punctured surface with puncture set $\mathcal{M}$, and let $\mathsf{B}$ be the $\mathbb{Z}[q^{\pm1/2}]$-basis of the Roger–Yang skein algebra…

Let $S$ be an oriented surface with negative Euler characteristic, let $\varphi$ be a periodic diffeomorphism, and let $[\gamma]$ be a $\varphi$-invariant smooth character with…

Let $\mathord{\mathbb{S}}$ be an orientable surface, and let the multicurve basis be the basis of the surface skein algebra given by multicurves. Applying Chebyshev polynomials of…

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 $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ punctures, let $\mathcal{C}(\Sigma_{g,n})$ be its curve algebra, let $\mathcal{C}(\Sigma_{g,n})'$ be the subalgebra generated…

Let $\Sigma_{g,n}$ be a punctured surface, let $\mathcal{C}(\Sigma_{g,n})_{\mathbb{C}}$ be its complexified curve algebra, and let $\mathcal{T}^{d}(\Sigma_{g,n})$ be its decorated…

Band–triangular basis conjecture. The (quantized) band basis coincides with the common triangular basis after localization at the frozen variables.

Fix a $n^2$-root $\omega=q^{1/n^2}\in\mathbb{C}-\{0\}$. For each ideal triangulation $\lambda$ of the punctured surface $\mathfrak{S}$, let $\mathscr{S}^q_n(\mathfrak{S})$ be the s…

Let $\Sigma$ be an unpunctured surface, let $\Delta$ be an ideal triangulation of $\Sigma$, and for each internal edge $E\in e_{\mathrm{int}}(\Delta)$ let…

Let $\mathsf{BWeb}_{\mathfrak{sl}_{3},{\Sigma}}$ be the graphical basis of the quantum $\mathfrak{sl}_3$ skein algebra of $\Sigma$, and let cluster monomials be monomials formed fr…

Let $\mathbb{I}^{\rm br}$ and $\mathbb{I}$ denote the conjectural bracelets and bangles basis maps, respectively, and let their structure constants be defined by multiplication in…

Let $\mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^{T})$ be the tropical-integer lamination set, and let $\mathbb{I}^{\rm br}$ be the map obtained by applying the paper's constructi…

Let $\mathscr{A}_{\rm L}(\frak{S};\mathbb{Z})$ be the set of ${\rm SL}_3$ laminations and let $\mathbb{I}^{\rm br}_{{\rm SL}_3}$ be the map obtained by replacing loop components wi…

Let $\Sigma_g$ be a closed genus-$g$ surface and $H_g$ a genus-$g$ handlebody. The reduced skein module $\mathcal{S}_q^{\mathrm{red}}(H_g)$ carries the representation of the skein…