8 problems

Let $S$ be a finite-type orientable surface with finitely many marked points, or vertices, on its boundary and in its interior, and let $\mathcal{A}(S)$ be its arc complex. The pur…

Finite-type conjecture. Every arc complex $Arc(F)$ is of some finite type.

Finite-type conjecture. Each arc complex of a connected bordered surface determines an underlying space of finite type $n<\finite$.

Sphericity conjecture. Provided $N$ is non-negative, the arc complex $Arc(F)$ is PL-homeomorphic to the sphere of dimension $N$.

Sphericity conjecture. The space $Arc(F)$ is piecewise-linearly homeomorphic to the sphere

An arc complex is the simplicial complex whose simplices encode collections of compatible arcs on a surface with boundary. Sphericity conjecture. The full arc complex of a surface…

Schleimer's conjecture. There is a constant $t(M)$ such that the monodromy of every fibered knot $K\subset M$ has translation distance in the arc complex of the fiber at most…

Let $S$ be a surface with a set of marked points $P$, let $C$ be a relevant boundary component, and let $\alpha,\beta\in\mathcal{A}_{C}(S,P)$ be arcs. Write…