12 problems

Let $W$ be a genus-two handlebody embedded in a compact, orientable 3-manifold $M$, and let $\alpha$ and $\beta$ be refilling meridians of $W$. Denote the corresponding refilled ma…

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 $M$ be a $3$-manifold, let $W\subset M$ be a genus two handlebody, and let $\alpha$ and $\beta$ be meridian disks of $W$. The exterior $\overline{M-W}$ is irreducible, and…

Let $M$ be a closed $3$-manifold obtained by Dehn filling a $1$-cusped $3$-manifold $N$ with a $1$-efficient ideal triangulation $\mathcal{T}$. Applying the Gang–Yonekura formula d…

Let $M$ be a closed $3$-manifold obtained by Dehn filling a $1$-cusped $3$-manifold along a slope $\alpha$, and let $\mathcal{I}_M(\alpha)$ denote the resulting closed-manifold ind…

Let $M$ be a $1$-cusped hyperbolic $3$-manifold. A rational-real-part cusp shape means that the cusp shape has rational real part for some choice of basis $\mathbf{m},\mathbf{l}$ o…

Let $M$ be a $1$-cusped hyperbolic $3$-manifold with cusp shape $c_{1}$. Cusp-shape norm conjecture. If … for sufficiently large $|p|+|q|$, then … The conjecture proposes that, asy…

Let $M$ be a $1$-cusped hyperbolic $3$-manifold, and let $M(p,q)$ denote the manifold obtained by $(p,q)$-Dehn filling on its cusp. Write $N_M(v)$ for the number of Dehn fillings o…

C. McA. Gordon's conjecture. Suppose that $M$ is a hyperbolic knot manifold $M$ and $\alpha, \beta$ are slopes on $\partial M$ such that $M(\alpha)$ is small Seifert and $M(\beta)$…

C. McA. Gordon's conjecture. For any hyperbolic knot manifold $M$, we have $\#\mathcal{E}(M) \leq 10$ and $\Delta(\mathcal{E}(M)) \leq 8$. Moreover, if $M \ne M_1,M_2,M_3,M_4$, the…

Let $N$ be a simple 3-manifold with boundary component of genus two, and let $a$ and $b$ be basic curves on that boundary. Write $N[a]$ and $N[b]$ for the manifolds obtained by att…