114 problems

Let $\mathcal{H}_g^{V(r)}$ be the small Heisenberg homology, let $\mathcal{H}_g^{V,\dagger}$ be the direct sum of the standard twisted homology groups, and 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…

For every knot $K\subset S^3$, let $F_K^N(x,q)=F_K^{\mathfrak{sl}_N,\mathrm{sym}}(x,q)$ denote the series associated with symmetric representations, let $\Delta_K(x)$ be the Alexan…

Let $M$ be a hyperbolic $3$-manifold with a regular triangulation $\mathcal{T}$ and NeumannZagier datum, including a choice of quad type with $\det \mathbf{B}\neq 0$. Let…

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

mod $p$ non-vanishing conjecture. The function $\widetilde{\tau }_p(M)$ is nowhere zero on $\mathbb{F}_p^\times$. The source says this is supported by computer calculations and giv…

$p$-adic non-vanishing conjecture. The function $\tau ^p(M)$ is nowhere zero on $U(\mathcal{O}_p)$. The source presents this as a generalization of the root-of-unity non-vanishing…

Let $M$ be an integral homology sphere, let $\zeta _8$ be a primitive eighth root of unity, and define $\widetilde{\tau }_8(M)=\sqrt{-1}^{\lambda(M)}\tau _8(M)$. Eighth-invariant n…

Let $M$ be an integral homology sphere and let $\tau _\zeta(M)$ denote its Witten–Reshetikhin–Turaev invariant at a root of unity $\zeta$. Non-vanishing conjecture. For every root…

Let $X$ be a Seifert manifold with base $S^2$, and let $(l,\underline{n}')\in\mathcal I_2^b$. Suppose that there do not exist elements $g_1,\ldots,g_n\in\operatorname{SU}(2)$ satis…

Let $X$ be a closed oriented $3$-manifold and let $G=\operatorname{SU}(2)$. Let $\mathcal M_j$ be the union of the connected components of the moduli space of flat $G$-connections…

Real Volume Conjecture. For any $(W,L,\rho)$,

Let $(W,L,\rho)$ be a hyperbolic-like triple, and let ${\mathcal T}_{{\mathcal I}}$ be an ${\mathcal I}$-triangulation of $(W,L,\rho)$. Denote by $c_{{\mathcal I}}(W,L,\rho)$ the c…

Circulator trace conjecture. In the case of stable Gelfand–Kazhdan categories,

Let $\mathcal{T}$ be a triangulation, let $R$ be the coefficient ring, and let … be the surjective $R$-module homomorphism from the quantum gluing module to the Panitch–Park quantu…

Let $\Sigma_{g,n}$ be a compact oriented surface, and let … be its mapping class group, with homeomorphisms and isotopies fixed on the boundary. For an odd integer $r\geq 3$, let…

Let $K$ be a hyperbolic knot, let $D$ be an open diagram of $K$, and let $\rho$ be a decorated representation of $D$ that is geometrically nondegenerate, meaning that $\rho$ is non…

Let $K\subset{\mathbb S}^3$ be a hyperbolic knot, let $\gamma$ be a simple closed curve on $\partial\nu(K)$, and let $\rho_0$ be the discrete faithful representation associated wit…

Let $M\cong N/\pi_1(M)$ be a geometric integer homology sphere with model geometry $N$, and let $W_M(\xi)$ be its properly normalized Witten–Reshetikhin–Turaev invariant for a prim…

Let $K$ be a knot with a well-defined GM series, and let $C_{(tp,sp)}(K)$ denote its $(tp,sp)$-cabling with $\gcd(s,t)=1$. Cabling Formula. The cabling formula above holds for any…

Let $K$ be a knot and $P$ a pattern whose pattern-defining link $P'$ has a well-defined GM series. Denote the satellite by $C_P(K)$. Satellite Formula. The GM series of $C_P(K)$ is…

Let $L$ be a link with a well-defined GM series, and let $L'$ be obtained by partial $-1/r$ surgery on an unknot component, or by $\infty$-surgery in the corresponding case. Partia…

Let $L$ be a link with a well-defined GM series … and let $Q$ be its linking matrix with zero diagonal. General link wavefunction conjecture. The wavefunction in…

Let $K_1$ and $K_2$ be knots with well-defined GM series, and let … be the manifold obtained by gluing with $\gamma=\begin{pmatrix}r&a\p&b\end{pmatrix}$. Write the GM series as…

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