29 problems

An orientable ribbon graph is a graph embedded in an orientable surface, and its partial-dual polynomial records the genera of all partial duals, with coefficients counting partial…

Let $\Gamma$ be a connected ribbon graph. Denote its bigraded $n$-color homology by $CH_n^{i,j}(\Gamma;\mathbb{C})$, where $i$ is the homological grading and $j$ is the quantum gra…

Căldăraru's conjecture. For every genus $g\geq 2$, there is a natural identification

Conjecture on dual cohomology classes. The dual cohomology classes of $W_{(m_a),n}$ can be expressed as

Let $\mathcal{M}_{g,P}$ be the moduli space of smooth genus-$g$ curves with marked points $P$, and let $W_{m_*,P}$ denote the combinatorial class associated with ribbon graphs havi…

Kontsevich's conjecture. This functional equals $\widetilde{\mathcal M}_{m_*}^{n}$ for a class $x$ of the form

Witten's conjecture. The combinatorial cycles can be expressed in terms of Mumford-Morita-Miller classes.

Moffatt's conjecture. Ribbon graphs are well-quasi-ordered under the ribbon graph minor relation.

Let $G$ be a connected graph and let $\Gamma$ be a ribbon graph associated with $G$. Suppose that $G$ has a bridge. Consider the spectral sequence of $n$-color homologies of…

Let $G$ be a bridgeless connected graph and let $\Gamma$ be a signed ribbon diagram of $G$. For $n$ a positive integer, write $T(\Gamma,n,t)$ for the total face color polynomial. T…

Let $\Omega(\overline{\textsf N}^{fr})$ be the cobar construction of the properad of framed curves, let $\Omega(\textsf{Frob}^*)$ be the cobar construction of the dual Frobenius pr…

Let $d$ be an integer, and let $\textsf{OGC}_{2d+1}^{{\:\raisebox{3pt}{\text{\circle*{1.5}}}}}$ be the oriented graph complex and…

Let $G$ be a ribbon graph, and let $~^{\partial}{\mathscr{E}}^{*}_{G}(z)$ denote its partial-$*$ polynomial. A polynomial is odd or even when all terms with nonzero coefficients ha…

Baker and Wang's conjecture. There exists a vertex $q$ of $G$ such that

Baker–Wang conjecture. The torsors $b_v$ and $r_v$ agree for all vertices $v$ if and only if $G$ is planar.

Baldridge's conjecture. The homology theory could be used to count the number of Tait colorings of the argument graph.

Căldăraru's conjecture. The complex $(\mathsf{RGC}_0,\delta+\Delta)$ computes the compactly supported cohomology of ${\rM}_g$.

Căldăraru's conjecture. The twisted complex $(\mathsf{RGC}_0,\delta+[m_2,-])$ computes the compactly supported cohomology of ${\rM}_g$, up to a degree shift.

Compatibility conjecture. The diagrams comparing the Chan–Galatius–Payne map with the maps induced by $F$ and by the map from hairy oriented graph complexes to ribbon graph complex…

Let $H(\mathrm{O}\,\mathrm{GC}_1,\delta_0)$ denote the homology of the oriented fixed-target graph complex with differential $\delta_0$, and let $H_c(\mathcal{M}_{g,n},\mathbb{Q})$…

Baker–Wang's conjecture. For any non-planar ribbon graph, there exists a base point such that the rotor-routing action and the Bernardi action with the given base point are differe…

A regular map is a graph embedded in a surface with the regularity property that its automorphism group acts transitively on incident vertex-edge-face flags. A regular map is self-…

Let $\cM_{g,n}^{\blacktriangle,\rm disc}$ denote the subspace of the decorated moduli space obtained by imposing the discrete-homotopy condition on connected metric ribbon graphs,…

Let $(G,\rho)$ be a ribbon graph. Write $V(G)$ for its vertices, $E(G)$ for its edges, $\operatorname{Pic}^0(G)$ for its degree-zero Picard group, and $\mathcal{T}(G)\subset E(G)$…

For $N\geq 1$, let $\mathcal R_N$ be the ring of formal series in $\lambda_1,\ldots,\lambda_N$ whose homogeneous components are rational functions, and let $f_N^o(\Lambda,s,s_-)$ b…