19 problems

A link is alternative if it belongs to the class of alternative links, and pseudoalternating if it belongs to the class of pseudoalternating links introduced by Murasugi and Maylan…

Limsup volume conjecture. For every non-splittable link $L$,

For each prime $p$, let $T_p=((\mathbb{Z}_p)^2,\lambda_pJ)$ be a symplectic quandle, where $\lambda_p\in\mathbb{Z}_p^\times$ may depend on the link crossing number. Let $H$ be the…

Let $n$ be a positive integer, and let $\{\mathfrak{n},\ldots,1,0\}$ be a sequence. Link spectrum realization conjecture. Any sequence $\{\mathfrak{n},\ldots,1,0\}$ satisfying a se…

Asymptotic average genus conjecture.

Let $L$ be a link, and let $\widetilde{\operatorname{Kh}}(L)$ denote its reduced Khovanov homology, graded by the … -gradings. This rules out a possible exceptional support pattern…

Let $L$ be the four-component link L8n8, whose linking diagram is a cycle, and let $\operatorname{I}^\natural(L)$ denote its instanton Floer homology. L8n8 conjecture. The link L8n…

Let $K$ be a link with a diagram $D(K)$, and let $G_{D(K)}$ be its associated graph. Denote by $\operatorname{hn_{cc}}$ the hull number associated to the cycle-convexity interval f…

Detcherry–Kalfagianni–Yang colored-Jones volume question.

Detcherry–Kalfagianni–Yang generalized Turaev–Viro volume conjecture.

Let $L$ be a link in a contact three-manifold $Y$, and let $\alpha$ be any contact form on $Y$ adapted to $L$. Denote by $ECK(L,Y,\alpha)$ and $\widehat{ECK}(L,Y,\alpha)$ the full…

Let $L$ be a non-trivial, non-split link, and let $E(L)$ denote its exterior. A closed surface $F$ containing $L$ has each component of $L$ non-separating in $F$ when no component…

Let $L$ be an amphicheiral link in $S^3$, meaning that an orientation-reversing homeomorphism of $S^3$ maps each component of $L$ to a component of $L$, possibly reversing orientat…

Let $L$ be a prime, nonsplit two-component link, and let $\Sigma(L)$ denote its intrinsic symmetry group. The link has full symmetry when $\Sigma(L)=\Gamma_2$, where $\Gamma_2$ is…

Alternating-link conformality conjecture. All subdivision rules arising from alternating links in the cited work satisfy the 1,2,3-tile criterion and are conformal.

Conformality conjecture. The subdivision rules for all alternating links are conformal.

Let $m,p_1,dots,p_r$ satisfy … for all $k,j=1,\dots,r$. Consider the sets $\mathcal{T}^{m+1}(\vee_{k=1}^r S^{p_k+1})$ and $\mathcal{H}_+(m+1;p_1+1,\dots,p_r+1)$, related by the cor…

so(3) volume conjecture. For all knots and links $L$,

Completeness conjecture. The class of monotonic moves is complete for computing canonical forms and hence for deciding isotopy.