46 problems

Let $D_1$ and $D_2$ be reduced alternating diagrams of prime links. They are Reidemeister equivalent if one can be transformed into the other by a finite sequence of Reidemeister m…

Let $D$ be a reduced alternating diagram of a knot $K$ with determinant $p$, where $p$ is prime. A Fox $p$-coloring is a coloring of the arcs of $D$ by elements of $\mathbb Z_p$ sa…

Let $L$ be a link. The tail of the colored Jones polynomial of $L$, if it exists, is a power series $\Phi_L(q)$ whose first $N$ coefficients agree, up to a common sign, with the fi…

Polynomial-time conjecture. Alternating link equivalence can be decided by a deterministic algorithm whose running time is polynomial in the size of the input diagrams.

Exponential decay conjecture. Both proportions exponentially decay with respect to crossing number:

Sakuma–Weeks conjecture. Crossing arcs of a reduced alternating diagram are the arcs of the canonical cell decomposition of the link complement.

Let $L$ be an alternating link, and write its Conway-normalized Alexander polynomial as … where the coefficients have alternating signs and no internal zeros. Fox's conjecture. The…

Let $K$ be an alternating hyperbolic link in $S^3$. Its hyperbolic volume is denoted by $\operatorname{vol}(K)=\operatorname{vol}(S^3\setminus K)$, and its determinant by $\det(K)$…

Greene's conjecture. If a pair of links have homeomorphic branched double-covers, then either both are alternating or both are non-alternating.

Murasugi–Przytycki's conjecture. For every alternating diagram $D$, this inequality is exact:

Let $L$ be an alternating link, and let $\mathcal{H}(L)$ be its bigraded homology theory. The Jones polynomial and the signature of $L$ are the data under consideration. Alternatin…

Second Tait conjecture. Any reduced alternating diagram of a link is minimal.

Let $L$ be a link with $c$ components. Call $L$ Khovanov-basic if its Khovanov complex $\operatorname{Kh}(L)$ is homotopy equivalent to a direct sum of basic complexes and exactly…

For a link diagram $D$, a crossing arc is a segment perpendicular to the projection plane that connects the upper strand of a crossing to the lower strand of the same crossing. Thi…

H-thin implies T-thin conjecture. Every H-thin link is T-thin. In particular, every non-split alternating link is T-thin.

Bar-Natan–Garoufalidis–Khovanov conjecture. Every non-split alternating link is homologically thin and not homologically slim.

Defect proportion comparison conjecture. For all even $c\geq 4$ and all odd $c\geq 11$, the proportion of $c$-crossing prime alternating links $L$ with $\gamma(L)>\Gamma(L)$ is lar…

Crosscap-three classification conjecture. For each even crossing number $2k\geq 6$, there are exactly

2d log-concavity conjecture. The absolute values $(|a_{ij}|)$ form a 2d log-concave sequence with no interior zeros. This conjecture is proposed as a bidimensional analogue of log-…

Strong version of Ishii's conjecture. The coefficients satisfy $a_{ij}=(-1)^{\mu+i+j+1}|a_{ij}|$ for all $i,j$, and $(a_{ij})$ has no interior zeros. The source notes that this str…

Let $L$ be an alternating hyperbolic link, with hyperbolic volume $\operatorname{vol}(L)$ and determinant $\operatorname{det}(L)$. Volume–determinant conjecture. … The conjecture h…

Let $L$ be a prime link with reduced alternating diagram and determinant $b\deltad$. A Fox $b\deltad$-coloring assigns colors to the arcs of the diagram. Generalized Kauffman–Harar…

Greene–Owens cubiquity conjecture. $\Sigma(L)$ bounds a rational homology $4$-ball if and only if $\Lambda(L)$ is cubiquitous.

Let $\mathcal{L}$ be a $\mathbb{Z}^2$-periodic alternating link, with toroidally alternating quotient link $L$. Let $P(z,w)$ be the characteristic polynomial of the toroidal dimer…

Two-minima one-saddle conjecture. If $L$ bounds a ribbon surface with two minima and a single saddle point, then $L$ is algorithmically ribbon.