17 problems

Let $L \subset \Sigma \times I$ be a link, where $\Sigma$ is a surface. An essential surface for $L$ is a surface of smallest genus in which $L$ can be embedded in the thickened su…

Let $M$ be a smooth oriented $4$-manifold for which Donaldson's polynomials are defined and non-zero, and let $\Sigma\subset M$ be a smoothly embedded oriented surface with positiv…

Let $C$ be a smooth curve of genus $g$ in a complex surface $X$, and let $K$ be a canonical class of $X$. Its adjunction relation is … For any smooth embedded $2$-manifold $\Sigma$…

Let $\mathcal K(l,m,n)$ denote the Klein bottles constructed in the paper. An embedded surface is ribbon when it admits a ribbon presentation, equivalently one with only ribbon sin…

Let $\mathcal K(l,m,n)$ denote the Klein bottle constructed from the parameters $l,m,n$. Its meridians are said to have order 4 when they represent elements of order $4$ in the fun…

Let $AP$ be the four-manifold constructed in the paper, and let $H_2(AP;\mathbb Z)$ denote its second homology group. An embedded surface in $AP$ is homologically essential if its…

Let $K$ be an Alexander polynomial one knot in $S^3$, and let $\Sigma\subseteq D^4$ be a locally flat, embedded, compact, orientable surface with boundary $K$ and complement fundam…

Let $\Sigma\subseteq S^4$ be a locally flat, embedded, closed, oriented surface. It is unknotted if it bounds a locally flat, embedded handlebody in $S^4$. Unknotting conjecture fo…

Sharpness conjecture. There exist $X$, $\Sigma$, and $\Sigma'$ as above such that

Let $\Sigma$ be a surface in the cube tessellation, and let $T^n$ be the quotient torus formed from the tessellation. Call cubes of edge length $2$ large cubes. Assume that each of…

Let four rays in $\mathbb{R}^3$ emanate from the same vertex, and assume that they do not lie in a half-plane. An order one end is an end asymptotic, outside a compact set, to a tu…

Embeddedness conjecture. This end is a complete, properly embedded end of type $(m,n,m+n-1)$. The surrounding discussion says that numerical evidence supports embeddedness, but tha…

Embeddedness conjecture. If $\phi\ne-\psi$ and $\phi\ne\pi-\psi$, then the surface is embedded. The conjecture is motivated by the observation that the level curves in the planes…

Let $K\subset {\mathbb{R}}^N$ be an embedded torus with one boundary component, and suppose that $\partial K$ is a unit circle. A closed curve in $K$ is non-null-homotopic if it is…

Let $K$ be a surface of genus $g$ embedded in ${\mathbb{R}}^N$ by a bilipschitz map, and let $\gamma_1,\dots,\gamma_{3g-3}$ be the boundary curves of a pants decomposition of $K$.…

Let $X$ be a simply-connected $4$-manifold, and let $B$ be an embedded surface in $X$ of genus $g_B$ with self-intersection … Assume that the class $[B]$ is divisible by $2$. Embed…

Generalized Thom conjecture. The genus $g_{\omega}(A)$ is less than or equal to the genus of any smoothly embedded connected surface representing $A$.