30 problems

Let $T_{3n-2,3n-2}$ be the $(3n-2)\times(3n-2)$ torus grid with its induced $l^1$ metric, and let $\mathrm{VR}(T_{3n-2,3n-2};n)$ be the Vietoris–Rips complex at scale $n$. Write…

Method-failure conjecture. The method also fails for all $h\geq 8$.

Homeomorphic reconstruction conjecture. The reconstructed dynamics might be homeomorphic to the original dynamics at a lower dimension than that needed for a diffeomorphically corr…

For a knot type $K$, let $n_{\operatorname{SC}}(K)$ denote its simple cubic birth level, and let $G_{n_{\operatorname{SC}}(K)}^{\operatorname{SC},\operatorname{BFACF}}(K)$ be the c…

Let $K$ be an amphichiral knot type and let $\omega$ be a minimal simple cubic lattice representative of $K$. Consider the minimal BFACF graph and the reflected mirror seed of…

Mirror-pair conjecture. These $56$ knotoids form $28$ mirror pairs. This is supported by the observed mirror-pair structure of the final reduction, but the source presents it as a…

Let $K_i$ be a knotoid diagram, and let $\operatorname{Reid}_r(K_i)$ denote the set of diagrams obtained by systematically performing Reidemeister moves with $r\geq 0$ crossing-inc…

Unpinning avoidance game complexity conjecture. The problem $\mathsf{Unpin_{AVOID}}(\Sigma,s)$ is $\textsf{PSPACE}$-complete for any $s$ and $\Sigma$, and…

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 $L$ be a physically realizable computation and let $h(L)$ denote its homological complexity. Homological Church–Turing thesis. Every physically realizable computation has finit…

Let $f$ be a function, let … and let $h(L_f)$ denote the homological complexity of the inversion problem associated with $f$. Homological characterization of one-way functions. The…

Let $L$ be a computational problem, let $h(L)$ be its ordinary homological complexity, and let $h_q(L)$ be a proposed quantum homological complexity measure. Quantum homological co…

Let $\mathcal{C}$ be a natural complexity class, let $h(L_n)$ denote the homological complexity of the restriction $L_n$ of a problem $L$ to inputs of size $n$, and let…

Let $\mathcal{C}$ be a natural complexity class, let $h(L)$ denote the homological complexity of a problem $L$, and let $S_{\mathcal{C}}$ be a subset of $\mathbb{N}\cup\{\infty\}$.…

Let $L$ be a decision problem in the bounded-error quantum polynomial-time class $\mathcal{BQP}$, and let $h(L)$ denote its homological complexity. Quantum homological obstruction…

Let $L$ be a computational problem with homological complexity $h(L)$. A physical system is said to solve $L$ efficiently when it computes solutions to $L$ within the relevant effi…

Let $L$ be a computational problem, let $T_L(n)$ denote its time complexity, and let $h(L)$ denote its homological complexity. Homological Time Complexity Relation. There exists a…

3-sphere conjecture. For a countable family of $(n,k)$ pairs, potentially including the pairs listed above, $\mathrm{VR}(T_{n,n};k)$ is homotopy equivalent to $S^3$. The pair…

Linear-size conjecture. The filtration $\mathcal{D}_{\infty}^{1}(X)$ consists of $O(kn)$ simplices.

Let $k$ be the parameter in the algorithmic result referred to as Theorem … precludes W[1]-hardness. The statement is presented as a conjectural implication in the context of param…

Linear void-complexity conjecture. Similar bounds should hold for the translates of a convex shape that is not necessarily a polytope.

Let an appropriate higher-dimensional analogue of a merge tree be given, and consider the authors' algorithm for finding cancellations of critical cells using generalized merge tre…

Let $X$ be a $c_1$-digital image, and let $dH_q(X)$ and $H_q^{c_1}(X)$ denote, respectively, its digital cubical singular homology and its $c_1$-cubical homology in degree $q$. Iso…

Adaptive-radius homology computation conjecture. One can compute the homology of