11 problems

Kinoshita-type conjecture. Every $P^2$-knot in $S^4$ is of Kinoshita type.

Method-failure conjecture. The method also fails for all $h\geq 8$.

Unknottedness conjecture. Every $fZ_2$-surface in $S^4$ is topologically unknotted.

Let $\Sigma\subseteq S^4$ be a locally flat, embedded, closed, orientable surface of genus $g$, and suppose that $\pi_1(S^4\setminus\Sigma)\cong\mathbb{Z}$ is generated by a meridi…

Let $X^4$ be a smooth $4$-manifold equipped with a trisection, and let $\Sigma$ and $\Sigma'$ be isotopic embedded surfaces in $X^4$. Consider bridge trisections of $\Sigma$ and…

Chart simplification conjecture.

Let $(F,\Gamma)$ be a branched covering surface-knot of degree $N$. Let $u_w(F,\Gamma)$ and $u(F,\Gamma)$ denote the weak simplification number and simplification number, respectiv…

Let $(F,\Gamma)$ be a branched covering surface-knot. The quantities $u_w(F,\Gamma)$ and $u(F,\Gamma)$ denote, respectively, the weak simplification number and simplification numbe…

Let $F$ be an orientable surface, let $\widetilde F$ be its universal cover, and let $c:[0,1]\to F$ be a closed smooth curve whose lift to $\widetilde F$ is injective. Call $c$ pre…

Let $F \subset \mathbf{R}^4$ be an orientable surface knot. Here $w(F)$ denotes the width of $F$, and $\operatorname{Braid}(F)$ denotes its braid index. Width bound conjecture. The…

Let $\tilde{R}_3$ be the symmetric dihedral quandle, and let $H_3^{Q,\rho}(\tilde{R}_3,\mathbb{Z})_Y$ denote its third symmetric quandle homology group with integer coefficients an…