103 problems

Witten's finiteness conjecture. The vector space $Sk(M)$ is finite dimensional over $\mathbb{Q}(A)$.

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 closed, oriented $3$-manifold, let $G$ be a reductive algebraic group, and let $\operatorname{Sk}_{G}(M)$ denote the vector space spanned by framed labelled graphs in…

Let $M$ be a closed $3$-manifold, let $S(M)$ denote its Kauffman bracket skein module, and let $HP^{\bullet}_{\#}(M)$ be the Abouzaid–Manolescu homology. Abouzaid–Manolescu dimensi…

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 $G$ be a reductive group, let ${}^{L}G$ be its Langlands dual group, and let $Y$ be a closed $3$-manifold. The associated $G$- and ${}^{L}G$-skein modules are vector spaces who…

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…

Przytycki's torsion conjecture. The following are equivalent:

Let $M$ be a closed oriented $3$-manifold, and let $\mathcal S_{2,\infty}(M)$ denote its Kauffman bracket skein module over $\mathbb Z[A^{\pm 1}]$. Marché's decomposition conjectur…

Strong finiteness conjecture. There exists a finite collection $Sigma_1,\dots,Sigma_k$ of essential subsurfaces $Sigma_i\subsetpartial M$ such that, for each $i$, the dimension of…

Let $M$ be a submanifold of a rational homology sphere. Assume that $M$ contains no closed, oriented incompressible surface. Let ${\cal S}_3(M)$ denote its Homflypt skein module, l…

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…

Quotient-module conjecture. The module ${\mathcal S}_3(M)$ is always a quotient module of

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

Alternating-sign polygon reduction conjecture. Any $2n$-gon of sign type $(+,-,+,\ldots,-)$ can be expanded as a sum of webs consisting of polygons of smaller sizes and $2n$-gons o…

Let $M$ be the quaternionic manifold, obtained as the quotient of $S^3$ by the action of the quaternionic group $Q_8$, and let $S(M;\mathbb{Z}[A^{\pm1}],A)$ denote its Kauffman bra…

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…

Let $M$ be a $3$-manifold, let $K(M)$ denote its Kauffman bracket skein module, and let $\overline{K(M)}$ be its completion. The element $1+t$ determines the torsion under consider…

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…