11 problems

Let a tri-plane diagram be a diagram describing a bridge trisection of an embedded $S^2$ in $S^4$, and count its total crossings across the three constituent plane diagrams. Low-cr…

A $2$-knot is an embedded $S^2$ in $S^4$, and its tri-plane crossing number is the minimum total number of crossings in a tri-plane diagram representing it. The spun trefoil and th…

Let the tri-plane crossing number of a $2$-knot be the minimum total number of crossings among its tri-plane diagrams. The spun trefoil is the $2$-knot obtained by spinning the tre…

Let $(A)_d$, $(B)_d$, $(C)_d$, and $(D)_d$ be hexagonal lattice diagrams, and let the corresponding bridge trisections be those determined by these diagrams. Deperturbation conject…

Shadow-diagram move conjecture. The diagrams $\mathfrak D^1$ and $\mathfrak D^2$ can be related by a finite sequence of moves, each of which is one of the listed types.

Bridge-trisection stabilization conjecture. There are diffeomorphic bridge trisections $\mathbb T_1'$ and $\mathbb T_2'$ such that, for $\varepsilon\in\{1,2\}$,…

Classification conjecture. Every $(2,0;2)$-bridge trisection is diffeomorphic to one of those described in Example and Figure.

Classification conjecture. There are exactly three, up to mirroring, totally irreducible $(2,0;1)$-bridge trisections.

Efficient bridge trisection conjecture. If

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…

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…