23 problems

Let $\Sigma\subset S^4$ be an oriented knotted surface. Let $\mathcal{G}=(G,E,\partial,\triangleright)$ be a finite crossed module. Write $I_\mathcal{G}(S^4\setminus\nu(\Sigma))$ f…

Amplitude invariant conjecture. For every 2-knot $K$ and every CRS movie $M$ representing $K$,

Cyclic knot group conjecture. Every locally flat surface in $S^4$ with cyclic knot group is unknotted.

Let $T^p$ be a $p$-dimensional torus embedded in $S^{p+2}$. Consider extendable homeomorphisms of the embedding, and distinguish their mapping classes in $\operatorname{MCG}(T^p)$…

Let $F_g$ be a closed orientable surface of genus $g$ embedded in $S^4$. An orientation-preserving self-homeomorphism of $F_g$ is extendable if it extends to a homeomorphism of the…

Let $B^{4}$ be the smooth $4$-ball. For an exotically knotted pair of closed surfaces $(\Sigma_{1},\Sigma_{2})$ in $B^{4}$, let $d(\Sigma_{1},\Sigma_{2})$ denote the least nonnegat…

Upsilon-move conjecture. Except for some very specific cases, the Upsilon moves are not induced by the usual welded moves. Consequently, they provide a large class of nontrivial mo…

The $RC_k$-equivalence conjecture. Two ribbon tubes are $RC_k$-equivalent if and only if they cannot be distinguished by finite type invariants of degree $<k$.

Let a $2$-knot be a surface-knot in $S^4$, and let its $3$-dimensional cross-section be the knot obtained by intersecting it with an appropriate $3$-sphere. Cross-section conjectur…

Let $F$ be an embedded $2$-sphere in $S^4$. The 4-dimensional unknotting conjecture. The sphere $F$ bounds a $4$-ball if and only if the fundamental group of its complement is infi…

Let $S$ be a properly embedded smooth surface in $B^4$. Genus-zero bridge trisection conjecture. Every properly embedded smooth surface $S$ in $B^4$ has a genus $0$ bridge trisecti…

Let $T(2,p)$ denote the $(2,p)$-torus knot, and let $K_p$ be its 2-twist spin. A knot is a root of the ribbon concordance graph if no distinct knot ribbon-concords to it from below…

Let $(X,\mathcal K)$ be a pair consisting of a 4-manifold and a knotted surface. A generalized bridge trisection is a generalized bridge decomposition of this pair, and two such tr…

Satoh's conjecture. Equivalence classes of w-knots are in bijective correspondence with ribbon torus knots in $\mathds{R}^4$.

A virtual broken surface diagram (VBSD) is a broken surface diagram for a virtual $2$-link. A graph change is a diagram change preserving the Gauss code of the double point curves…

Let $KI$ be the Kishino virtual knot, and let $S_T(F\times [0,1],K)=(F\times S^1\times [0,1],K\times S^1)$ be the generalized spinning construction. Non-realizability conjecture. T…

Let $\mathcal{K}$ be a knotted surface in $S^4$ admitting a $(b;b-1,c_2,c_3)$--bridge trisection. Such a bridge trisection is a surface of the indicated type, with $b$ the bridge p…

Let $\mathcal{K}$ be a knotted surface in $S^4$, and let $S^4\setminus\mathcal{K}$ be its exterior. The fundamental group of the exterior is $\pi_1(S^4\setminus\mathcal{K})$. Cycli…

Let $T$ be the $3$-twist-spun trefoil, viewed as a $2$-knot, and let $C$ be a cross-cap embedded in $4$-space. Write $T\mathbin{\#}C$ for their connected sum, and let…

Functoriality conjecture. Link Floer homology is functorial with respect to marked smooth isotopy classes of surfaces embedded in four-space: a marked surface with boundary…

A graph change of a virtual broken surface diagram is a modification preserving the Gauss code of the double-point curves that either passes a double-point curve through another cu…

A virtual broken surface diagram (VBSD) is a broken surface diagram for a virtual $2$-link; a graph change is a modification that preserves the Gauss code of the double-point curve…

Let $KI$ be the Kishino virtual knot, and let $S_T(KI)$ denote its spun virtual $2$-torus obtained by the construction $S_T(F\times [0,1],K)=(F\times S^1\times [0,1],K\times S^1)$.…