43 problems

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…

Let $L$ be a link. Its unlinking number $u(L)$ is the minimum number of crossing changes needed to transform $L$ into an unlink, and $u_{BJ}(L)$ denotes its Bernhard–Jablan unlinki…

Let $kge 2$, let $\beta\in \mathfrak{Br}_n$ be a braid, and let $L(\beta)$ denote its link closure. Write $\mathrm{H}_{0|k}(\beta)$ for the homology constructed in the paper 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…

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

Let $\widehat{\sigma}$ be the link associated to a braid $\sigma$, and let $Kh^{\sharp,k}(\widehat{\sigma})$ denote the degree-$k$ part of the higher-dimensional Floer homology, wh…

Ball number conjecture. For any nontrivial and nonsplittable link $L$,

Let $T$ be an oriented 1-tangle diagram in the annulus, and let $L_{T^0}$ be the link in $S^2 \times S^1$ obtained by closing it with an overpass or underpass arc. Let…

Let $L\subset S^3$ be a framed oriented link with multivariable Alexander polynomial $\nabla_L$, and suppose $\nabla_L$ is nonzero. Let $F_L^t$ be the regularized formal series ass…

Let $L$ be a framed link in $S^3$ with $V$ components, let $B$ be its linking matrix, and let $L_{\alpha}$ denote the link with labels…

Improved ball-number conjecture. For every link $L$,

The link surgery conjecture. Whenever the right-hand side makes sense, for some $\epsilon\in\{\pm1\}$ and $d\in\mathbb{Q}$,

Let $L$ be an $l$-component link, let $J_L(n_1,\ldots,n_l;q)$ be its colored Jones polynomials, and let $\nabla_L(x_1,\ldots,x_l)$ be its Alexander–Conway function. The link GM-ser…

Let $L$ and $L'$ be mutant links, and let $T$ be a tangle satisfying the hypotheses of the paper's mutation-invariance theorem for delta-graded knot Floer homology. Positive-mutati…