107 problems

Let $K$ be a knot. The unreduced Khovanov homology $Kh(K;\mathbb{Z})$ is the Khovanov homology of $K$ with integer coefficients, and $2$-torsion means an element of order $2$. Shum…

Let $\mathcal{C}$ be the classical knot concordance group. An element of $\mathcal{C}$ is 2-torsion if it has order two, and a knot is negative amphichiral if it is isotopic to its…

Let $K$ be a number field, let $n=[K:\mathbb{Q}]$, and let $\epsilon>0$. Write $Cl(K)[l]$ for the subgroup of elements of the class group $Cl(K)$ annihilated by $l$, and let $D_K$…

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…

Strong uniform boundedness conjecture. For fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on $\#({}^{\phi}L)_{\operatorname{tors}}$ as $L$ ranges over extensions 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 loopless graph, and let $H^{i,*}_{{\mathcal A}_m}(G)$ denote its graph cohomology associated with ${\mathcal A}_m$. A triangle is a cycle of length $3$, and a square i…

Let ${\mathcal A}_m$ be the algebra used to define graph cohomology, let $P_v$ be the polygon with $v$ vertices, and let $H^{i,j}_{{\mathcal A}_m}(P_v)$ denote its bigraded cohomol…

Let ${\mathcal A}_m$ be the algebra used to define graph cohomology, let $P_v$ be the polygon with $v$ vertices, and let $H^{i,j}_{{\mathcal A}_m}(P_v)$ denote its bigraded cohomol…

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 $i\geq m-2$.

Boundary-range conjecture. There is some $m_0$ such that, if $n_0=2m_0-6$ or $n_0=2m_0-7$, then $\widetilde H_{\nu_{m_0,n_0}}(M_{m_0,n_0})$ is an elementary $3$-group.

Exponent-three conjecture. The exponent in Theorem $$ is $3$.

The $n=14$ conjecture. The result holds for $n=14$ as well.

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.