27 problems

For every annular link $L$, the annular Khovanov homology of $L$ is nonzero in the $k$-grading equal to the wrapping number of $L$:…

Let two spheres in $\mathbb{R}^3$ be equipped with oriented normal bundles and intersect along a circle in the space $\tilde Y$, with the canonical orientation induced on the inter…

Let ${\mathcal X}_{\kappa}$ be the character variety with parameter $\kappa$, and let ${{\mathcal E}(\rho)}$ be the set of end invariants of a character $\rho$. The ending invarian…

Let $\rho$ be an $\mathrm{SL}(2,\mathbb C)$ character and let ${{\mathcal E}(\rho)}$ denote its set of end invariants, viewed as a subset of the projective lamination space…

Neumann's surjectivity conjecture. If $k=1$, then $\mathbb{U}$ is onto. In general, every $\mathsf{PSL}(2,\mathbb{R})$-representation with dense image lies in…

Let $M$ be an orientable, irreducible closed $3$-manifold with infinite fundamental group. Virtual fibering conjecture. $M$ has a finite cover that fibers over the circle. This is…

Let $M$ be an orientable, irreducible closed $3$-manifold with infinite fundamental group. Virtual infinite Betti number conjecture. Either $\pi_1(M)$ is virtually solvable or $M$…

Let $M$ be an orientable, irreducible closed $3$-manifold with infinite fundamental group. Virtual positive Betti number conjecture. Some finite cover of $M$ has positive first Bet…

Let $M$ be an orientable, irreducible closed $3$-manifold with infinite fundamental group. Waldhausen's virtual Haken conjecture. $M$ is finitely covered by a Haken manifold. This…

Rudolph's characterization conjecture. A ribbon is quasipositive if and only if, for some $n$, it is ambient isotopic to a subribbon of $\widetilde T_{n,n}$. This would characteriz…

Andrews–Curtis conjecture. Any two contractible simple polyhedra can be related by a sequence of pentagon, hexagon, and $2$-$0$ moves and their inverses, ignoring all gleams.

Let $S$ be a rational homology projective plane with quotient singularities, and write … The smooth locus $S^0$ is assumed to be simply connected. Kollár's algebraic Montgomery–Yan…

Let $Y$ be a closed oriented 3-manifold satisfying … If $Y$ is not $S^3$, then Kirby's $SU(2)$ representation conjecture. there exists a homomorphism … with non-abelian image. This…

Let $Y$ be a toroidal rational homology $3$-sphere. The refined toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 6$, then $Y$ is not $SU(2)$-abelian. This strengthen…

Let $Y$ be a toroidal rational homology $3$-sphere. The toroidal small-homology conjecture. If $|H_1(Y;\mathbb{Z})| \le 4$, then $Y$ is not $SU(2)$-abelian. This conjecture predict…

Let $N$ be a positive integer, let $T^2$ be the torus, and let $\mathrm{SkCat}_{\mathrm{SL}_N}(T^2)$ and $\mathrm{SkAlg}_{\mathrm{SL}_{N/d}}$ denote the corresponding skein categor…

Let $A_\Gamma$ be a right-angled Artin group associated to a finite simple graph $\Gamma$. For a second homology class $\alpha\in H_2(A_\Gamma)$, let $M_\alpha$ denote the associat…

Let $K$) be a knot, let $D(K)$ denote its positively-clasped Whitehead double, and let $[K]$ and $[D(K)]$ denote their elements in the knot concordance group $\mathcal{C}$. Kirby’s…

Kronheimer–Mrowka's well-definedness conjecture. The quantity $J^\flat(F)$ is well-defined: it does not depend on the choices made in the seam-vertex cancellation step.

Let $\Sigma$ be the surface under consideration, let $\bar{\Gamma}(\Sigma)$ be the extended mapping class group, and let $\mathcal{A}(\Gamma(\Sigma))$ be the quotient algebra defin…

Trisection-genus-three classification conjecture. Every irreducible 4-manifold with trisection genus three is either the spin of a lens space, or a Gluck twist on a specific 2-knot…

SU(2) representation conjecture. If $Y\not\cong S^3$, then there is a nontrivial homomorphism

Furuta–Ohta–Seiberg–Witten correspondence conjecture.

PL surface projection moves conjecture. The moves for singularities of projection of a PL surface should be the analogues of the moves for the singularities of projection of a smoo…

Let $M$ be a finite-volume hyperbolic $3$-manifold. A $3$-manifold is Haken if it contains a properly embedded incompressible surface. Virtual Haken conjecture. The manifold $M$ ha…