65 problems

Hougardy–Lutz–Zelke conjecture. Every triangulation of an orientable surface of genus $g$ with

Let $S_{g,p}$ be a genus-$g$ surface with $p$ punctures, let $\brun(S_{g,p})$ denote the set of Brunnian pseudo-Anosov mapping classes on $S_{g,p}$, and let $L(\brun(S_{g,p}))$ den…

Let $F$ be a closed surface, let $x_i\in F\times\{0\}$ for each $i$, and let ${\cal S}_{2,\infty}(F\times I,\{x_i\}_1^{2n};R,A)$ denote the relative Kauffman bracket skein module w…

Stable torsion length conjecture. If

Let $S$ be an orientable surface, let $M$ be a 3-manifold, and let $f\colon S\to M$ be a map. The induced homomorphism is … Simple Loop Conjecture. If $f_*$ is not injective, then…

Let $S$ be a surface, and let $\operatorname{Convex}(S)$ denote the space of convex curve functionals on $S$, equipped with the product topology inherited from its values on the cu…

Non-orientable surface complexity conjecture. Both $\mathsf{Pin}(\Sigma,s)$ and $\mathsf{SimplePin}(\Sigma,s)$ are in $\textsf{P}$ when $s\leq 3$, and are $\textsf{NP}$-complete wh…

Hardness-starts-at-four conjecture. For a fixed orientable surface $\Sigma$, the problem $\mathsf{SimplePin}(\Sigma,4)$ is $\textsf{NP}$-complete.

Let two rainbow diagrams represent isotopic surfaces. A rainbow moves conjecture. Any two rainbow diagrams representing isotopic surfaces are related by a sequence of rainbow moves…

Let $\Sigma_g$ be a closed oriented surface of genus $g$, and let $\gamma,\gamma'$ be nonseparating simple curves that are homologous and disjoint from pairwise disjoint curves…

Let $S$ be an infinite-type surface properly homotopy equivalent to a graph $\G$, and let $\widetilde{\Theta}:\operatorname{ExMap}(S)\to\operatorname{Map}(\G)$ be the continuous in…

The decorated-tiling fruitfulness conjecture. Any purely noncommutative decorated tiling of a polygon provides a fruitful seed. More generally, a decorated tiling of an arbitrary s…

Multi-arc smoothing lower-bound conjecture. The multi-arc $calpha$ has a smoothing that cannot be homotoped to a multi-arc with fewer than

Random-curve taut smoothing conjecture. A random curve has a taut smoothing.

Let $\Sigma$ be a compact, connected, orientable surface, and let $\mathcal A$ be a multiarc and curve graph on $\Sigma$. For a witness $W$, let…

Let $\Sigma$ be a compact, connected, orientable surface, and let $\mathcal A$ be a multiarc and curve graph on $\Sigma$. A compact, essential subsurface $W\subset\Sigma$ is a witn…

Let $\Sigma$ be a punctured surface with puncture set $\mathcal{M}$, and let $\mathsf{B}$ be the $\mathbb{Z}[q^{\pm1/2}]$-basis of the Roger–Yang skein algebra…

Let $\mathord{\mathbb{S}}$ be an orientable surface, and let the multicurve basis be the basis of the surface skein algebra given by multicurves. Applying Chebyshev polynomials of…

Let $\tilde{G}$ and $\tilde{G}'$ be tangled tree diagrams related by crossing changes, and let $G\in\mathsf{Tree}_{\mathfrak{sp}_4,\Sigma}$ be a tree-type elementary web. Crossing-…

Let $S$ be a surface, and let an object be naturally associated to $S$ and have sufficiently rich structure. Ivanov's metaconjecture. Every such object has the extended mapping cla…

Let $\Sigma$ be a compact hyperbolic surface, and let $I_k(\Sigma)$ denote the maximal self-intersection number of a $k$-systole on $\Sigma$. Erlandsson–Parlier's conjecture. One h…

Let $\mathbb{S}=S_{g,h}$ be a closed surface notation used in the paper, let $k$ be the rank, and let $\mathbb{E}_{\rm good}$ map to the dosp mutation graph via the map…

Let $\Sigma$ be an orientable surface with finitely many punctures and no boundary, and let $\mathcal{P}(\Sigma)$ denote its deformation space of complex projective structures. Eac…

Let $S$ be a closed orientable surface with a constant-curvature metric. Consider the space of geodesic triangulations of $S$ and the subgroup of isometries of $S$ homotopic to the…

Let $\Sigma$ be an unpunctured surface, let $\Delta$ be an ideal triangulation of $\Sigma$, and for each internal edge $E\in e_{\mathrm{int}}(\Delta)$ let…