22 problems

Let $\mathsf{\Gamma}$ be a finite connected graph, and let $\mathbb{B}_n(\mathsf{\Gamma})=\pi_1(UC_n(\mathsf{\Gamma}))$ denote its graph $n$-braid group. A right-angled Artin group…

Let $\mathcal{G}_{(2)}$ be the specified subclass of graphs from the paper, and let $UD_2(\mathsf{\Gamma})$ denote the discrete unordered configuration space of two points on a gra…

Let $\mathsf{\Gamma}$ be a graph, let $i$ be a homological degree, and let $\lambda$ be a partition. Write $\mathcal{H}_i(\mathsf{\Gamma})$ for the relevant representation-valued h…

Let $\mathcal{G}$ be a collection of graphs. Say that $\mathcal{G}$ is a set of universal generators in degree $i$ if, for every graph $\mathsf{\Gamma}$, the group…

Let $\mathsf{\Gamma}$ be a graph and let $k\geq 0$. Write $F_k(\mathsf{\Gamma})$ for its $k$-point configuration space. Folklore torsion-freeness conjecture. The space…

For a graph $G$, let $F_n(G)$ be the configuration space of $n$ ordered points in $G$, and let $cmathcal{TGI}$ be the category of topological graph embeddings. The assignment…

Let $F_n(G)$ be the configuration space of $n$ ordered points in a graph $G$. For a class of graphs, universal finite generation in degree $i$ and with $n$ particles means that a f…

Consider the trijunction $\Gamma_T$ and the simplified anyon models whose symbols have at most four labels. Let $N_0(\Gamma_T)$ denote the particle number beyond which all consiste…

Let $\Gamma$ be a graph, and let $N_0(\Gamma)$ be a number such that the consistency relations for all particle numbers $N>N_0(\Gamma)$ are satisfied by solutions of the consistenc…

A graph-braided anyon fusion model assigns braiding exchange operators to graph-braid generators and satisfies polygon equations expressing compatibility between fusion and braidin…

Universal generation conjecture. For any planar graph $\mathsf{\Gamma}$ and $i>0$, the $\mathbb{Z}[E]$-module $H_i(B(\mathsf{\Gamma}))$ is generated by classes arising from disjoin…

Finite generation of LCS quotients. Every successive quotient of the lower central series,

Let $G$ be a graph which is neither a line segment nor a circle. For each $i$, let $b_i(n)$ denote the $i$th Betti number of the graph braid group configuration space: … Recall tha…

Let $\Gamma$ be a graph and let $B_n\Gamma$ denote its braid group on $n$ strands. A graph is an $n$-nucleus if $B_n\Gamma$ is not a right-angled Artin group, while $B_n\Gamma'$ is…

Let $\Gamma$ be a planar graph, let $B_n\Gamma$ be its braid group, and let $P_3\Gamma$ be its pure braid group on three points. A group is simple-commutator-related if it has a pr…

Let $P_2\Gamma$ be the pure braid group of two points on a planar graph $\Gamma$. A group is simple-commutator-related if it has a presentation whose relators are commutators. Abra…

Let $B_2\Gamma$ be the braid group of two points on a planar graph $\Gamma$. A group is simple-commutator-related if it has a presentation whose relators are commutators. Farley–Sa…

Let $D_2\Gamma$ be the discrete configuration space of two ordered points on a graph $\Gamma$, let $P_2\Gamma=\pi_1(D_2\Gamma)$ be the pure graph braid group, and let $\beta_1$ den…

Let $B_n\Gamma=\pi_1(UC_n\Gamma)$ be the braid group of $n$ points on a graph $\Gamma$, and let $H_1(B_n\Gamma)$ denote its first homology group. Kim–Ko–Park's conjecture. The grou…

Let $B_2\Gamma$ be the braid group of two points on a planar graph $\Gamma$. A group is simple-commutator-related if it has a presentation whose relators are commutators. Farley–Sa…

Commutator-relator characterization conjecture. If ${\mathrm{\bf B}}(G,n)$ has a presentation in which every relator is a commutator, then $G$ can be constructed as in Lemma.

Planar graph braid group commutator presentation conjecture. There exists a presentation for $B_2\Gamma$ such that the relators are all commutators corresponding to pairs of disjoi…