35 problems

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…

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…

Asymptotic torus-link hull number conjecture. For every $p$ there exists an integer $n$ such that, whenever $q>n$, the $(p,q)$-torus link has hull number $p$.

Galashin--Lam's conjecture. Determine the curves $C$ for which

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…

Polynomial-growth conjecture. The total dimension satisfies

Symmetry conjecture. For $\ast=\boldsymbol{\mathcal B}$, the corresponding matrix elements agree under $\gamma_{\ast}^n$, namely the path sums

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…

Hogancamp–Rose–Wedrich's conjecture. The intrinsically colored homology of positive torus links is concentrated in even $t$-degree.

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…

Consider refined Chern-Simons theory with gauge group $\mathrm{SU}(N)$ and torus links, with colors labeled by rectangular Young diagrams. Let the large-$N$ limit refer to the limi…

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 $M$ and $N$ be positive integers, let $k=\gcd(M,N)$, and set $m=M/k$ and $n=N/k$. Let $L_{M,N}$ be the symmetric function defined as the weighted sum over the relevant tuples o…

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 ${\bm{b}}=(b_1,\ldots,b_m)$ and $N=\sum_{i=1}^m b_i$. Let $T(m,m;{\bm{b}})$ be the ${\bm{b}}$-colored $(m,m)$-torus link, and let $E_{\bm{b}}$ be the algebra defined from the c…

Positive torus links are known to enjoy a parity property, meaning that their colored triply-graded homology has the parity behavior discussed in the paper. Colored torus-link pari…

Let $K$ be an algebraic knot in $\mathbb{RP}^3$ of degree $d$, let $N_d=(d-1)(d-2)/2$, and let $T_{\mathrm{proj}}(d,d-2)$ denote the quotient of the $(d,d-2)$-torus link in…

Let $K$ be a smooth, irreducible real algebraic curve in $\mathbb P^3$ of degree $d\geq 3$, let $N_d=(d-1)(d-2)/2$, and suppose $K$ is an $MW$-curve with $w(K)=N_d$. The projective…

Let $T_{m,n}$ be a torus link with $L$ components. For fixed representations $\rho_1,\ldots,\rho_L$, genus $g$, and degree $\beta$, let…

Let $T_{m,n}$ be a torus link with $L=\operatorname{gcd}(m,n)$ components. Let $\mathcal{M}_{g,h,\beta}$ be the relevant moduli space of holomorphic curves, and let…

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…

Recursive Khovanov homology conjecture. Except for the following cases,