59 problems

Let $T_n$ be a trivial link of $n$ components, and let $L_0,L_1,L_2,L_3$ be a standard unoriented skein quadruple, with $L_{i+1}$ obtained from $L_i$ by a right-handed half-twist o…

Let $M$ be a simply connected $3$-manifold, possibly with holes, and let $S_{2,\infty}(M)$ denote its Kauffman bracket skein module at parameter $2$ and infinity. Traczyk's conject…

Handlebody basis conjecture. For every handlebody $H_n$, ${\mathcal S}_3(H_n)$ is the free $Z[x_0^{\mp 1},x_1^{\mp 1},x_2^{\mp 1}]$-module with basis

Let $M=M_1\# M_2$, where each $M_i$ is a 3-manifold, possibly with holes, and neither $M_i$ is equal to $S^3$. Let ${\cal S}_{2,\infty}(M)$ denote the Kauffman bracket skein module…

Let $F$ be a closed surface, let $x_i\in F\times\{0\}$ for each $i$, and let ${\cal S}_{2,\infty}(F\times I,\{x_i\}_1^{2n};R,A)$ denote the relative Kauffman bracket skein module w…

Let $M$ be an integral homology sphere, let $K(M)$ denote its Kauffman bracket skein module, and let $\overline{K(M)}$ be its completion. Homology-sphere $(1+t)$-torsion conjecture…

Comparison conjectures. (a) The projection

Geometric deformation conjecture. (a) For each oriented $3$-manifold, the epimorphism

Linear-independence conjecture. The trivial links $T_i$ are linearly independent in $\mathcal{S}_4(S^3)$ in either of the following cases:

Fourth skein module generation conjecture. The fourth skein module $\mathcal{S}_4(S^3)$ is generated by trivial links; the fourth skein module of $2$-tangles in a disk is generated…

Montesinos–Nakanishi conjecture. Every link is $3$-equivalent to a trivial link. More generally, every $2$-tangle is $3$-equivalent to a tangle with at most one crossing; every $3$…

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…

Safronov's root-of-unity skein conjecture. There is a line bundle $\mathcal{L}_q$ on $\mathrm{Loc}_{G^\vee}(M)$ such that

Skein–DT sheaf conjecture. There is an isomorphism

Quantum-group abelianization conjecture. The abelianization map $\mathrm{Ab}$ is equivalent to the abelianization map constructed using quantum groups.

Flow-loop/BPS-series conjecture. For every fibered link $K$,

Let $M$ be a closed, oriented $3$-manifold, and let $\mathscr S(M)$ denote its Kauffman bracket skein module over $\mathbb Z[A^{\pm1}]$. A module decomposition into free modules an…

Trivial-character skein-module conjecture. If $X(M)$ is finite, then

GJS conjecture.

Let $M$ be a closed, oriented $3$-manifold, and write its skein partition function as an Euler expansion with integer exponents $c_k$. Positivity conjecture for skein partition fun…

Let $L$ be a link in the solid torus. Its wrapping number is the minimum geometric intersection number of $L$ with a meridional disk of the solid torus. Cubic skein determination o…

Let $[n_1,n_2,\dots,n_r]$ and $\epsilon[n_1,n_2,\dots,n_r]$ be two diagrams of rational links, where $\epsilon=1$ if $r$ is odd and $\epsilon=-1$ otherwise. The Rational Tangle Alg…

Let $q\in\mathbb{C}^\times$ be a quantum parameter, and consider the finiteness and holonomicity assertions for skein modules stated in Theorems and. Non-root-of-unity extension co…

Let $R=Q(q^{\frac{1}{2}})$, let $L\subset S^3$ be a link with $n$ components, and let $E_L$ denote its exterior. For each component, let $t_i\in\mathcal{S}(E_L)$ be the meridian of…

Let $G$ be a reductive group, $G^{L}$ its Langlands dual, and $M$ a closed, connected, oriented $3$-manifold. Let $\operatorname{Sk}^{\operatorname{gen}}_{G}(M)$ denote the skein m…