15 problems

Let $M$ be a closed $3$-manifold obtained by Dehn filling a $1$-cusped $3$-manifold $N$ with a $1$-efficient ideal triangulation $\mathcal{T}$. Applying the Gang–Yonekura formula d…

Let $M$ be a closed $3$-manifold obtained by Dehn filling a $1$-cusped $3$-manifold along a slope $\alpha$, and let $\mathcal{I}_M(\alpha)$ denote the resulting closed-manifold ind…

Let $p,q>0$ be coprime integers with $pq eq 0pmod{2}$, and let $mathcal{T}{(p,q)}$ be a $1$-efficient ideal triangulation of the complement of the $(p,q)$ torus knot. The alternati…

Operator-insertion conjecture. For every $\mathcal O\in\mathcal M(\mathcal T)$: (a) there exists a linear $q$-difference operator $\widehat{A}_{\mathcal T,\mathcal O}$ with a funda…

Holomorphic-block factorization conjecture. For every 1-efficient triangulation $\mathcal T$ there exists a palindromic linear $q$-difference operator $\widehat{A}_{\mathcal T}$ of…

Holonomic-equation conjecture. The state-integral and the rotated 3D-index satisfy identical pairs of linear $q$-difference equations, equal to the homogeneous part of the linear…

Let $I^{\mathrm{rot}}(n,n')(q)$ be the rotated 3D-index, and suppose it satisfies a linear $q$-difference equation in the two variables $n$ and $n'$. Let $A$ denote the $A$-polynom…

Let $\widehat{\Phi}(h)$ be the matrix of completed formal asymptotic power series associated with boundary-parabolic $\operatorname{PSL}_2(\mathbbm C)$-representations, and let…

Let $I^{\mathrm{mer}}(0,0)$ be the meromorphic 3D-index of a hyperbolic knot, let $\sigma$ range over the relevant boundary-parabolic $\operatorname{PSL}_2(\mathbbm C)$-representat…

Let $\tau$ tend vertically to $0$, let $I^{\mathrm{rot}}(n,n')$ be the rotated 3D-index, and let $\sigma$ range over boundary-parabolic $\operatorname{PSL}_2(\mathbbm C)$-represent…

Let $K$ be a hyperbolic knot in $S^3$, and let $\widehat{\Phi}^{(\sigma_1)}_{K,n}$ and $\widehat{\Phi}^{(\sigma_2)}_{K,n}$ be the computable asymptotic series associated with the g…

Let $(A_\mu,A_\lambda)\in\mathbb{R}^2$ satisfy $|A_\mu|,|A_\lambda|<\frac{\pi}{2}$, and suppose the complete hyperbolic structure deforms to an incomplete structure with holonomy…

Let $M$ be a one-cusped hyperbolic 3-manifold, let $\hat M$ be its end compactification, and let $\mathcal{X}^{\mathrm{geo}}$ be the relevant set of conjugacy classes of geometric-…

Stokes matrix–rotated 3D-index conjecture. For every hyperbolic knot $K$,

3D-index entry conjecture. A distinguished entry of the Stokes automorphism matrix is the 3D-index of Dimofte–Gaiotto–Gukov.