18 problems

Let $T(rl,kl)$ be a torus link, where $r$ and $k$ are relatively prime, and let $n=n_1+\cdots+n_l$. Let $f_{\lambda^1,\dots,\lambda^l}(t,\nu)$ be the reformulated colored HOMFLY po…

Let $\mathcal{L} = \mathcal{L}_1 \sqcup \dots \sqcup \mathcal{L}_d$ be a link with $\mathcal{L}_1$ an unknot and suppose $p \in \mathcal{L}_1$. For colors $k_1,\dots,k_d$, let…

Let $T(2,q)$ denote a $(2,q)$-torus link, let $[T(2,q)]$ denote its link type, and let ${\operatorname{Rib}}([T(2,q)])$ denote its infimal folded ribbonlength. Uniform torus-link c…

Polynomial-growth conjecture. The total dimension satisfies

Gorsky–Oblomkov–Rasmussen conjecture. The stable Khovanov homology $\operatorname{Kh}(T(n,\infty))$ is the dual to the homology of the Koszul complex associated to the sequence…

Greedy matching conjecture. The greedy matching is a Morse matching for all torus braids, that is, the graphs

Torus-link/circular-group isomorphism conjecture. For all $n,m\in\mathbb{N}^*$, this morphism is an isomorphism. The claim generalises the previously established coprime case; the…

Let $R^{(2,2k)}$ be the germ of the plane curve singularity $y(y-x^k)=0$, let $R^{(2,2k)\oplus N}$ denote the direct sum of $N$ copies of this singularity, and let…

Let $R^{(2,2k)}$ be the germ of the plane curve singularity $y(y-x^k)=0$, and let $\widehat{N\!Z}_{R^{(2,2k)}}(t)$ denote its motivic Cohen–Lenstra zeta function. Here $k$ is a pos…

Let $K_n(t,q)$ denote the Poincaré polynomial of the Khovanov homology of $T(n+1,n)$, and let $L_n(t,q)$ denote that of $T(n,n)$. Let $C_n=\frac{1}{n+1}\binom{2n}{n}$ be the $n$th…

Let $D_{n,m}^i$ and $E_{n,m}^i$ be the diagrams defined in the paper, and consider the saddle cobordism … at the crossing corresponding to the last letter $\sigma_i$. Surjectivity…

Let $M$ and $N$ be positive integers, and define … Let $L_{M,N}$ be the symmetric function associated with the $M,N$-path tuples, let $\mathbf{Q}_{m,n}$ be the elliptic Hall algebr…

Let ${\bm{b}}=(b_1,\ldots,b_m)$ and let $T(m,m;{\bm{b}})$ be the ${\bm{b}}$-colored $(m,m)$-torus link. Colored torus-link parity conjecture. The link $T(m,m;{\bm{b}})$ is parity;…

Let $N$ and $M$ be positive integers, write $d=\frac{\gcd(N,M)}{}$, $N=dn$, and $M=dm$ with $n$ and $m$ coprime, and define … Let $P_{n,m}$ be the operator from the coprime case, a…

Let $T_{(m,n)}$ be the $(m,n)$-torus link in $S^3$, with $m<n$. Let $TV_r(S^3\setminus T_{(m,n)})$ be its Turaev–Viro invariant. Chen–Yang integrality conjecture. If $r$ is coprime…

Let $T_n=T(n,n)\subset S^3$ and let $v_{C,n}$ be the lattice point where the $C^{th}$ and $(C+1)^{th}$ Alexander-support hypercubes meet. Assume the Heegaard Floer link homology is…

Let $T_n\subset S^3$ be the $(n,n)$-torus link. Fix a Heegaard diagram $\mathcal{D}_1=(\Sigma,\mathbf{\alpha},\mathbf{\beta},z_1,\dots,z_n,w_1,\dots,w_n)$ and a permutation…

Let $T(2,k)$ be the $(2,k)$-torus link, let $T(2,-k)$ be its mirror image, and let $u(L)$ denote the unknotting number of a link $L$. For an odd integer $k$, consider the connected…