33 problems

Quantum A-polynomial annihilation conjecture. The series $F_K(x,q)$ is annihilated by the quantum $A$-polynomial:

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

Gukov's conjecture. The pair

Let $K$ be a knot, let its noncommutative A-ideal be the peripheral ideal associated with the knot, and let its colored Kauffman brackets be the sequence of colored Kauffman bracke…

AJ conjecture. The homogeneous difference equation of $J_{\mathcal{K}}(N)$ with respect to $N$ gives the A-polynomial of $\mathcal{K}$.

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…

Torus-knot boundary-slope conjecture. Nontrivial torus knots are the only knots in $S^3$ with two boundary slopes.

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 $I^{\mathrm{rot}}(n,n')(q)$ 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 $\widehat{A}$ be the noncommutative $q$-difference operator associated with a knot, and let $A$ be its classical $A$-polynomial. AJ-Conjecture. Specializing $q=1$ in the…

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…

Let $K$ be a knot, let $J_{K,n}(q)$ be its sequence of colored Jones polynomials, and let $\hat L$ and $\hat M$ act on sequences $f\colon\mathbb N\to\mathbb Q(q)$ by … Let…

$(a,t)$-deformed annihilation conjecture. The deformed series satisfies

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

Generalized colored superpolynomial conjecture. The superpolynomial colored by a symmetric representation has the form

A-polynomial and defining-curve specialization conjecture.

AJ conjecture. The specialization of the quantum A-polynomial at $q=1$ is the usual A-polynomial of $K$.

For a knot $K$, let $A_K(\lambda,\mu)$ denote its $A$-polynomial and let $Aug_K(\lambda,\mu)$ denote its augmentation polynomial; their vanishing loci define the corresponding char…

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 $A(L,M,q)$ denote the non-commutative $A$-polynomial of a knot, and let $A(L,M)$ denote its commutative $A$-polynomial. AJ conjecture. There is a polynomial $B(M)$ in $M$ such…

Let $K$ be a knot, let $A(K;x,y)$ be its $A$-polynomial, and let $J^{{\mathfrak{sl}}(2)}_{[r]}(K;q)$ be the colored Jones polynomial in color $[r]$. AJ conjecture. The $A$-polynomi…

Let $K$ be a knot. Denote by $A_K(M,L)$ its $A$-polynomial, by $\widetilde{\alpha_K}(t,M,L)$ the recurrence polynomial generating the localized recurrence ideal of the colored Jone…

A knot invariant is associated with a knot, and the A-polynomial is an algebraic invariant of a knot describing its character variety. Quantization of the A-polynomial produces int…