71 problems

Let $L$ be a link, let $c(L)$ denote its crossing number, and let $\mathrm{Rib}(L)$ denote its folded ribbonlength, defined as the infimum of the core length divided by the ribbon…

Let $L$ be a link of braid index $b$, and let $B_1$ and $B_2$ be braid diagrams of $L$, each with $b$ strands. The writhe of a braid diagram is the signed crossing number. Jones' c…

Let $\mathcal L$ be an $r$-component link, with components colored by representations $R_1,\ldots,R_r$, and let $f_{(R_1,R_2,\ldots,R_r)}(q,\lambda)$ denote its reformulated invari…

Let $T_n$ be a trivial link of $n$ components, and let $L_0,L_1,L_2,L_3$ be a standard unoriented skein quadruple, with $L_{i+1}$ obtained from $L_i$ by a right-handed half-twist o…

Normalization conjecture. The number of $\mathcal{G}$-colourings of $D$ can be normalized to give a link invariant.

Let $L$ be a link, and let $c$ and $c'$ be components of $L$. Denote by $\widetilde{E}^k(L,c)$ the reduced Szabó spectral sequence associated to $L$ with distinguished component…

Let $L$ be a boundary link, and let $\phi_n(L)$ denote the associated link invariant. Factorial integrality conjecture. For every boundary link $L$ and every $n\geq 1$, … The autho…

Let $W$ be the fixed-point open subset associated with the involution on $\mathcal{Y}_{m,\tau}$, and let ${{\mathbb{T}}_{\hat{{\bf \alpha}}}}$ and…

Let $T(p,q)$ be a torus knot of type $(p,q)$, where $p$ and $q$ are coprime integers. Let $\rho_{j,k}\colon \pi T(p,q)\to\operatorname{SO}(3)$ send the standard generators $a$ and…

Let $L\subset S^3$ be a link with $N$ components, let $\Delta_L(t_1,\ldots,t_N)$ be its Alexander polynomial, and let $\rho\colon \pi L\to \operatorname{SO}(3)$ be an abelian repre…

Let $L$ be an oriented link, and let $r(L)$ denote the link obtained from $L$ by rotation of its rotor. The Jin–Rolfsen conjecture. The Alexander polynomials of $L$ and $r(L)$ are…

Let $L$ be an oriented link, let $n$ be a positive integer, let $GLZ^n_L(\tau,t)$ denote the invariant associated with the quantum superalgebra $U_q[\operatorname{osp}(2|2n)]$, and…

Let $L$ be an oriented link, let $m,n$ be positive integers, let $LG^{m,n}_L(\tau,t)$ denote the Links–Gould invariant, and let $\Delta_L$ denote the Alexander polynomial. Generali…

Let $\kappa$ be an oriented link, represented as the closure of an $m$-stranded braid, and let $\mathit{Kh}_{\mathrm{symp}}^*(\kappa)$ denote the graded symplectic invariant define…

Let $\lk\in\mathbb{Z}$ be a linking number, and let $\mathbb{Z}^{(2)}_r(\lk)$ denote the group of integer-valued KL-type $r$ invariants of two-component links with linking number…

Linear-independence conjecture. The trivial links $T_i$ are linearly independent in $\mathcal{S}_4(S^3)$ in either of the following cases:

Let $L$ be a non-split link. Write $c(L)$ for its crossing number, $g_T(L)$ for its Turaev genus, and $v_L(t)$ for its Jones polynomial. Qazaqzeh–Chbili–Lowrance conjecture. The li…

Let $L$ be a homogeneous link. The slice Cromwell equality conjectures. (i) The slice Cromwell inequality … is an equality if and only if $L$ is positive. (ii) For a homogeneous di…

Let $L$ be a multi-component link. Let $J(L)$ denote its Jones polynomial, and let … be the multiset of pairwise linking numbers. Write $(J,\operatorname{LK})$ for the combined lin…

Let $L$ be a link, and let $\Phi^{(A^{(i)}, B^{(i)})}_{X_2}(L)$, for $i\in\{1,2,3,4,5\}$, denote the five quantum enhancement polynomials associated with the canonical two-element…

Let $\iota:{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes(m-1)})\rightarrow{\rm End}_{U_q(\mathfrak{so}_{2n+1})}(S^{\otimes m})$ be given by…

Whitehead-link conjecture. (a) $W_n$ is not $n$-quasi-isotopic to the unlink. (b) $W_n$ is not strongly $1$-quasi-isotopic to the unlink.

A cts-link is a singular link considered in the paper, and $\widetilde{\Upsilon}_{x,y,z}$ and $\widetilde{\Upsilon}_{x,y,0}$ are the corresponding link invariants, with the latter…

Let $D$ be an oriented diagram for an oriented link $L$. Let $\tau_D$ be Kashaev's symmetric matrix associated with $D$, let $w(D)$ be the writhe of $D$, and let $\tau_D[x]$ denote…

Let $T$ be a symplectic quandle, and let the Allen–Swenberg series be the infinite family of links considered in the source, with $H$ the connected sum of two Hopf links. The sympl…