16 problems

Handlebody basis conjecture. For every handlebody $H_n$, ${\mathcal S}_3(H_n)$ is the free $Z[x_0^{\mp 1},x_1^{\mp 1},x_2^{\mp 1}]$-module with basis

Let $(M,W)$ be an admissible pair, where $W$ is a genus two handlebody with meridian disks $\alpha$ and $\beta$, and suppose that $M-W$ is $\partial$-irreducible. Boundary-irreduci…

Let $(M,W)$ be an admissible pair, with $W$ a genus two handlebody and $\alpha,\beta$ the specified meridian disks of $W$; the source defines admissibility and alignment in the pre…

Let $H_g$ be a genus-$g$ handlebody, and let $\tau$ be a one-vertex triangulation of its boundary. A $[\tau]$-layered-triangulation is a layered-triangulation of $H_g$ extending…

Let $H$ be a handlebody and let $S$ be a compact surface in $\partial H$. Let $\mathcal{N}(\partial S)$ be the subgroup of $\pi_1(S)$ normally generated by the boundary curves of…

Let $H$ be a handlebody with invariant measured lamination $\boldsymbol{\Lambda}$, and let the growth rate refer to the associated irreducible automorphism. A lamination is tight w…

A ribbon surface is a ribbon surface carried as a $2$-dimensional sub-$2$-handlebody of a $4$-dimensional handlebody. A 1-isotopy is a $(1,2)$-deformation between ribbon surfaces,…

Boundary-determination conjecture. If $\partial X$ and $\partial X'$ are diffeomorphic, then $X$ and $X'$ are diffeomorphic.

Let $W$ and $W'$ be $4$-dimensional $2$-handlebodies. Two such handlebodies are $2$-equivalent when they are related by a finite sequence of isotopies, handle slides, and creations…

Let $H_1$ and $H_2$ be 3-dimensional genus-$g$ handlebodies embedded in $S^4$, for an integer $g\geq 0$, with common boundary … They are compressing curve equivalent if there exist…

Let $H_n$ $H_m$ denote the connected sum of handlebodies of genera $n$ and $m$, let $\mathcal{S}_{2,\infty}(H_2)$ be the Kauffman bracket skein module of the genus-two handlebody,…

Let $H^i_0$ and $H^i_1$ be the handlebodies referred to in the source's preceding remark, and let $g$ denote the relevant genus. Linear boundary-length conjecture. There exists…

For each $g\geq 0$, let $V_0,V_1\subset S^4$ be 3-dimensional genus-$g$ handlebodies with the same boundary, $\partial V_0=\partial V_1$. A simple closed curve in $\partial V_0$ is…

Let $\bfF$ denote the category of finitely generated free groups. Consider the functor … that assigns to each bottom tangle $T:m\to n$ the homomorphism $(i_T)_*:F_n\to F_m$. The fu…

Let $H_{g,n}(q)$ be the Hecke algebra associated with the handlebody braid group, let $G_{g,n}$ be the corresponding group, and let $\Sigma_{g,n}$ be the set of elements … where ea…

Let $W$ be a genus-two handlebody embedded in a 3-manifold $M$, and let $[?]$ and $[?]$ be essential discs in $W$ whose refills produce exteriors $M[\alpha]$ and $M[\beta]$. Suppos…