10 problems

Let ${\cal K}$ be a simplicial complex of type $\{3,4\}$, meaning that every $(d-2)$-face is contained in $3$ or $4$ faces of dimension $d-1$. Let $(P_1,\ldots,P_t)$ be the corresp…

Let $APrism_m$ be the $m$-parameter antiprism, and let $Prism(APrism_m)$ denote the prism over it. Write $z$ for the zigzag vector and $Int$ for intersection vectors. Prism-over-an…

Let $C_p$ and $C_q$ be polygons, and set … Write $z$ for the multiset of zigzag lengths and multiplicities, and $Int$ for intersection vectors. Product-of-polygons conjecture. For…

Let $d$ be the dimension parameter, let $\beta_{d-1}$ be the regular $(d-1)$-dimensional complex and let $\alpha_{d-1}$ denote the corresponding simplex; write $Pyr(\beta_{d-1})$ f…

Let ${\cal K}$ be a complex of odd dimension. A zigzag in ${\cal K}$ has a signature, recording the numbers of intersections of the zigzag with the relevant types of faces. Zero-si…

DDF's conjecture. For every even $n\geq 20$ with $n\ne 22$, there exists a tight $5_n$.

$4_n$ existence and zigzag conjecture. (i) A z-knotted $4_n$ exists if and only if $n\geq 30$ and $n\equiv 2\pmod 4$. (ii) A tight $4_n$ exists if and only if $n\geq 8$ and…

Symmetry-group conjecture. (i) $3_n(D_{2h})$ exists if and only if $\frac{n}{4}$ is even and $n\geq 16$; there are no tight $3_n(D_{2h})$. (ii) $3_n(D_{2d})$ exists if and only if…

Curvature-graph conjecture. The graph of curvatures of any tight graph of type $3_n$ is the graph in the first of those three cases.

DDF's conjecture. Every z-knotted $3$-valent plane graph has an odd number of edges of type I.