10 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 $\mathcal K(l,m,n)$ 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…

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

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…

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

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…