69 problems

Throughout, let $H^i(S)=H^i(S;\mathbb{Z})$ denote the integral cohomology of a finite group $S$. For a finite $p$-group $P$, let $e(P)$ be the exponent of…

Let $K_5$ be the complete graph on five vertices. Complete-graph $K_5$ conjecture. … The source presents this as a conjectural computational pattern and gives no resolution.

Let $K_4$ be the complete graph on four vertices. Complete-graph $K_4$ conjecture. For every $m\geq4$, … The paper reports verification for $m\leq14$, but does not establish the st…

Let $P_3|P_3$ denote the graph formed by gluing two triangles along an edge, and let $H^{i,j}_{{\mathcal A}_m}(G)$ denote the corresponding graph cohomology. Square-with-diagonal c…

For $n>4$, let $W_n$ be the wheel graph, let $W_n^{out}$ be obtained by deleting an edge from its polygonal part, and let $W_n^{in}$ be obtained by deleting a spike edge. Wheel and…

Let $K_n$ be the complete graph on $n$ vertices, and let $w^1_{{\mathcal A}_3}(K_n)$ denote the width of the torsion in $H^{1,*}_{{\mathcal A}_3}(K_n)$. Width conjecture for comple…

Let $W_n$ be the wheel graph, the cone over an $(n-1)$-gon, and let $w^1_{{\mathcal A}_3}(W_n)$ denote the width of the torsion in $H^{1,*}_{{\mathcal A}_3}(W_n)$. Width conjecture…

Let $G_{t,s^k}$ be the graph obtained by gluing a triangle and $k$ squares in a sequence along edges, and let $w^1_{{\mathcal A}_3}(G)$ denote the width of the torsion in…

Let $G_{t,s^k}$ be the graph obtained by gluing a triangle and $k$ squares in a sequence along edges, with $v=3+2k$ vertices. Torsion formula for $G_{t,s^k}$. … This is one of the…

Let $P_{t,k}=P_3|P_k$ be the graph with $k+1$ vertices obtained by gluing a triangle and a $k$-gon along an edge. Let $w^1_{{\mathcal A}_3}(G)$ be the width of the torsion in…

Let $G$ be a graph, and let $H^{**}_{{\mathcal A}_m}(G)$ denote its graph cohomology associated with the algebra ${\mathcal A}_m$. A graph is loopless if it has no loops, and a cyc…

The $D_n$ conjecture. There exists an integer $n_0\geq 8$ such that, if $n\geq n_0$, then $\widetilde H_{\nu_{2n}}(M(D_n))$ is an elementary $3$-group. Moreover, if $n\geq n_0$ and…

Freeness conjecture. The reduced homology group $\widetilde H_i(M_{m,n})$ is free if and only if $i\geq m-2$.

Rank–torsion conjecture. The knots $K_1$ and $K_2$ have the same ranks of the Khovanov homology if and only if they have the same torsion.

Reduced-torsion conjecture. The knot $K$ is T-rich if and only if its reduced Khovanov homology has torsion.

H-thin implies T-thin conjecture. Every H-thin link is T-thin. In particular, every non-split alternating link is T-thin.

Powers-of-two torsion conjecture. The Khovanov homology of every link has no torsion of order $p$ for any $p$ other than a power of $2$.

Torsion existence conjecture. Except for the trivial knot, the Hopf link, and their connected sums, the Khovanov homology of $L$ has $2$-torsion.

Let $H$ be a free abelian group, and let $L(H)$ and $L'(H)$ denote the free Lie algebra and free quasi-Lie algebra on $H$, respectively. For $l\geq 1$, write $L_l(H)$ and…

Let $c_j$ be the weights in the metric-affine action, and let Minkowski space be equipped with connections of the form … Here $R(x)$ denotes the curvature of the connection, and th…

Let $X$ be a simply-connected finite $CW$-complex. For a prime $p$ and $r\geq 1$, say that $X$ is mod-$p^r$ hyperbolic if the number of $p^r$-torsion summands in the homotopy group…

Uniform torsion-freeness conjecture. There exist polynomials $\psi(x),\phi(x)\in\mathbb{Z}[x]$ such that

Tightness conjecture. For every prime $p$, $\Gamma_{n,p}$ is tight:

Let $\mathcal{HI}_{g,p}^b$ be the handlebody Torelli group, and let $\tau(\mathcal{HI}_{g,p}^b)$ and $\sigma(\mathcal{HK}_{g,p}^b)$ denote the images of the Johnson and Birman--Cra…

Moore's conjecture. The following are equivalent: