17 problems

Let $K$ be a knot. Let $P_K^0(q,E,Q)$ be the factor of the $A_q$-polynomial corresponding to the inhomogeneous recursive relation, let $A_K(l,\alpha)$ be the $A$-polynomial, let…

Let $K$ be a knot. Denote by $A_K(l,\alpha)$ its $A$-polynomial, by $A_q(K)(l,\alpha^2)$ the specialization at $q=1$ of the $A_q$-polynomial, and let $\varepsilon$ be the evaluatio…

Gukov's conjecture. The pair

Let $K$ be a knot. Its colored Jones polynomial is $J_K(n)=J_n(K;q)$ and satisfies a nontrivial recurrence … where $c_j(q,q^n)\in\mathbb{Z}[q,q^n]$. Define operators $E$ and $Q$ by…

Let $M$ be a complete finite-volume once-cusped hyperbolic three-manifold. An ideal triangulation of $M$ is a triangulation whose vertices lie at the cusp, and an ideal triangulati…

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 $C\in\mathcal{G}_{\mathbb{Z}}$ have strongly detected boundary slopes $\mathcal{DS}_{C}$, and let $K=\operatorname{Sat}(P,C,f)$ be a satellite knot whose embedded pattern knot…

Let $K$ be a knot, let $\mathcal{M}_{K}$ denote its exterior, let $A_K$ be its $A$-polynomial, let $\mathrm{Vol}(\mathcal{M}_K)$ be the sum of the volumes of the hyperbolic pieces…

SO-type super-A-polynomial branch conjecture. At $t=-1$, the zero locus

A-polynomial and defining-curve specialization conjecture.

Strong AJ conjecture. Suppose $K$ is a knot in $S^3$. Then

AJ conjecture. For every knot $K$ in $S^3$, $\varepsilon(\alpha_K)$ is equal to $A_K(M,L)$, up to multiplication by a polynomial depending only on $M$.

Quantum A-polynomial conjecture. For any knot $K\subset S^3$,

For a knot $K\subset S^3$, let $J\subset \mathbb{C}[m^{\pm1},l^{\pm1}]$ be the kernel of the algebra map determined by the meridian and longitude, and let $A_K(m,l)$ be the $A$-pol…

Let $K\subset S^3$ be a knot, and let $\mathbf{A}_K(e^x,e^p,Q)$ denote the three-variable $Q$-deformed $A$-polynomial associated with the mirror construction for the resolved conif…

Sikora's conjecture.

Quantum invariant detection conjecture. Under these hypotheses, $\limsup_{r\rightarrow\infty}\frac{1}{r}\lvert Z_r(M,f)\rvert$ defines a seminorm on $K_{-1}(T)$, and its radical is…