33 problems

Let $b$ be an odd positive integer, let $cmathcal M_b$ be the class of 3-manifolds under consideration, and let $cmathcal Gamma_b$ be the subring of unified invariants whose evalua…

Suppose that $L$ is an oriented link in $S^3$ with $mu$ components. Write its averaged Jones polynomial at $t=1$ as … and set $\phi_n(L)=(-2)^{\mu}a_{n+\mu}(L)$. An algebraically s…

Ohtsuki's factorial integrality conjecture. For every $n\geq 1$ and every integral homology 3-sphere $M$,

Let $\delta_n(M)\in V_n$ be the framing correction for the invariant $I_n(M)$ of a homology $3$-sphere $M$ decorated with a framing or bundled bordism. Vanishing conjecture for the…

Let $M$ be an integer homology $3$-sphere, let $\widetilde{I}_n(M)$ be the decoration-independent unframed invariant obtained by subtracting the framing correction from $I_n(M)$, a…

Categorification conjecture. The object $\Theta(M)$ should have at least the properties specified subsequently in the source, categorifying the normalized WRT invariant…

The p-modular TQFT structure conjecture.

Let $K$ and $\theta$ denote generators of a Borel-type Hopf subalgebra associated with the simplicial construction, and let the relevant Kuperberg and Hennings invariants be constr…

Let $M$ be a closed $3$-manifold with $\beta_1(M)\geq 1$, let $p$ be an odd prime, let $\zeta_p$ be a primitive $p$-th root of unity, and let $\lambda_L(M)$ be the Lescop invariant…

Let ${\cal V}^{DF}$ be Donaldson's TQFT associated with moduli spaces of connections on a non-trivial $SO(3)$ bundle. For $x_1,x_2,\ldots$, define … Let ${\cal V}^{(D)}$ be the pol…

Let $I_{k,n,\varphi}^{FN}$ denote the higher $PSU(n)$ knot invariant of Frohman and Nicas, and let $\Delta^{(\gamma)}$ denote a higher irreducible Alexander Character. Positivity c…

Let the KKTL invariant denote the Kontsevich–Kuperberg–Thurston–Lescop invariant of an integral homology sphere, and let the LMO invariant denote the Le–Murakami–Ohtsuki invariant.…

Let $M$ be a closed $3$-manifold, let $H_1(M)$ denote its first homology group, and let $N$ be the symplectic-fermion parameter appearing in the Lyubashenko invariant…

Let a plumbed 3-manifold be represented by an arbitrary, possibly non-reduced, plumbing, and let the associated $(q,t)$-series be specialized by taking $t\rightarrow 1$ to obtain a…

Let $\hat{Z}_a(q,t)$ be the refined invariant associated with a plumbed 3-manifold and a $\operatorname{Spin}^c$ structure $a$, and suppose that the recovered series $\hat{Z}_a(q)$…

Let $M$ be a bordered 3-manifold with $n$ toroidal boundary components, equipped with a system of arcs $\mathscr{A}\subseteq M$. Let $\mathcal{X}(M,\mathscr{A})^{\mathcal{L}}$ deno…

Let $K$ be a knot, let $r$ be the surgery parameter, and write … Let … be the Ramanujan theta function. Regularized $+1/r$-surgery conjecture. When the $+1/r$-surgery formula of Gu…

Let $K$ be a knot and consider the $+1$-surgery formula of Gukov and Manolescu for the invariant $\hat{Z}$. Regularized $+1$-surgery conjecture. When the $+1$-surgery formula conve…

The universal radial-limit conjecture. The invariant $N_\ell(M^3,\omega)$ is conjectured to satisfy

Let $H$ be a ribbon and semisimple Hopf algebra, let $X$ be a closed oriented $3$-manifold, and let $\overline{X}$ denote $X$ with reversed orientation. Write $X\#\overline{X}$ for…

Let $H$ be a finite-dimensional Hopf algebra, let $D(H)$ denote its Drinfeld double, and let $X$ be a closed oriented $3$-manifold. The invariants $Z_{\text{Kup}}(X;H)$ and…

Let $M$ be a closed $3$-manifold, and let the spherical category used to define its Turaev–Viro invariant have Drinfeld center defining a Reshetikhin–Turaev invariant. Turaev–Virel…

Knot-decorated decomposition conjecture. The WRT invariant satisfies

Index-factorization conjecture. The index satisfies

Categorification conjecture. There are series $\widehat Z_b(q)$, convergent for $|q|<1$, such that