91 problems

Let $K$ be a hyperbolic knot in $S^3$. A realization assigns functions … for positive integers $N$, where $\mathsf{X}(K)$ parametrizes hyperbolic structures on the complement of…

Let $M$ be a hyperbolic $3$-manifold, either closed, cusped or with totally geodesic boundary, and let $\operatorname{Vol}(M)$ denote its volume. As $r$ varies over the odd integer…

Let $K$ be a hyperbolic knot or link, so that $S^3\setminus K$ admits a hyperbolic structure. Let $J_{N-1}(K)$ be the normalized colored Jones polynomial at $q=\exp(\pi i/N)$, and…

Crossing-number bound conjecture. The bound

Parameterized Volume Conjecture. There exists an open subset $\mathcal{O}_K\subset\mathbb{C}$ such that for every $u\in\mathcal{O}_K$,

Let $K$ be a knot in $S^3$. The geometric quantized space for the $SO(3)$-Witten–Chern–Simons theory of the torus is the alternating subspace … of holomorphic sections of…

Let $L$ be a hyperbolic link in $S^3$, let $J_{L,m}(t)$ be its $m$-colored Jones polynomial, and let $\operatorname{Vol}(S^3{\smallsetminus} L)$ be the hyperbolic volume of its com…

Let $K$ be a hyperbolic knot, meaning that its complement $S^3\setminus K$ possesses a complete hyperbolic structure. Let $J_N(K;q)$ be the $N$-dimensional colored Jones polynomial…

Limsup volume conjecture. For every non-splittable link $L$,

Volume conjecture. For every knot $K$,

Murakami's parametrized volume conjecture. The function $H(K,u)$ is analytic on some open subset of $\mathbb{C}$, and the volume function $V(K,u)$ is given by

Let $\mathcal{M}$ be a cusped hyperbolic $3$-manifold, and let $Z_\gamma(\mathcal{M})$ be the quantum invariant obtained from the Faddeev quantum dilogarithm. The parameter…

Let $K$ be a knot in the three-sphere $S^3$, and let $J_N(K;t)$ denote its colored Jones polynomial associated to the $N$-dimensional irreducible representation of…

Volume conjecture.

Let $N=\#_k S^2\times S^1$ and let $T$ be a framed link in $N$. Let $J_d(N,T)(t)$ be its $d$-colored Jones invariant, and let $\operatorname{Vol}(T)$ denote the hyperbolic volume o…

Let $K$ be a knot, and let $J_N(K;t)$ denote its colored Jones polynomial associated with the $N$-dimensional irreducible representation of $sl_2(\mathbb{C})$, normalized by…

For a knot $K$ in $S^3$, define the sequence … where $J_K(n)$ is the colored Jones polynomial. The eventual monotonicity conjecture. For every knot $K$ in $S^3$, the sequence $a_n$…

Let $K$ be a knot, and let Kashaev's invariant be evaluated at positive integer parameters. The volume conjecture. The asymptotic behavior of Kashaev's invariant should determine t…

Let $M$ be a cusped manifold, define … and let $H_N(M)$ denote the proposed quantum hyperbolic invariant. Here ${\rm CS}(M)$ and ${\rm Vol}(M)$ are respectively the metric Chern–Si…

Let $M$ be a cusped manifold, and let $(W_n,L_n,\rho_n)$ be a sequence of compact hyperbolic Dehn fillings of $M$ converging to $M$, with $L_n$ the link of short simple geodesics f…

Let $(W,L,\rho)$ consist of a compact oriented 3-manifold $W$, a link $L$ in $W$, and a $PSL(2,\b7C)$-character $\rho$; let $H_N(W,L,\rho)$ be its quantum hyperbolic invariant and…

Let $(W,L,\rho)$ be a triple consisting of a $3$-manifold, a link, and the relevant representation. Let $K_N(W,L,\rho)$ be its quantum hyperbolic invariant, and let…

Let $K$ be a knot, let $\langle K\rangle_N$ denote its Kashaev invariant, and define the extended colored Jones function at a reduced rational $h/k$ by … In particular,…

Let $M$ be a closed oriented $3$-manifold and let $\mathcal{L}$ be a framed hyperbolic link with $n$ components in $M$. For an odd integer $r=2N+1$ with $N\geq 1$, let…

Let $r$ be odd, let $L\subset S^3$ be a hyperbolic link with components $L_1,\ldots,L_l$, and let $M=L((p_1,q_1),\ldots,(p_l,q_l))$ be a sufficiently large Dehn filling, meaning th…