941 problems

For every annular link $L$, the annular Khovanov homology of $L$ is nonzero in the $k$-grading equal to the wrapping number of $L$:…

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 $K$ be a knot, let $\sg(K)$ denote its signature, let $H_1$ denote the first homology group of the double branched cover of $K$, and let $Q_K$ be the associated polynomial valu…

HOMFLY–PT–Kauffman factorisation criterion. If the HOMFLY–PT and Kauffman polynomials for a 2-component link $\mathcal{L}$ satisfy … .

Let $K$ be a knot, and let $c_n(K)$ denote its $n$-crossing number, the minimum number of $n$-fold crossings in a projection of $K$. Strict inequality conjecture. For all knots $K$…

Chi-ribbon closure conjecture. Then $L_1$ is an alternating link. In particular, $K_1$ is an alternating knot in the knot case.

Let $K \subset S^3$ be an alternating knot. Its Alexander polynomial is … Here a polynomial is log-concave when its coefficient sequence satisfies $a_{i-1}a_{i+1}\leq a_i^2$ for ev…

Let $\beta$ be a braid whose closure $L(\beta)$ has one connected component, written $\pi_0(L(\beta))=1$. Let $\mathrm{HH}_{m|m}(\beta)$ denote the corresponding homology, graded b…

Cabling Conjecture. If $K$ is a knot in $S^3$ which has a reducible surgery, then $K$ is a cabled knot and the reducing slope is given by the cabling annulus.

Let $G$ be a $\theta$-graph in $S^3$. A belt $B$ is an oriented annulus intersecting the edges of $G$ in intervals, and $G(B)$ denotes the spatial graph obtained by buckling $G$ al…

Torus-knot conjecture. If $K$ is a nontrivial $SU(2)$-averse knot, then $K$ is a torus knot.

Strong invertibility conjecture. Every L-space knot in $S^3$ is strongly invertible: there is an orientation-preserving involution of $S^3$ whose fixed-point set is a circle inters…

Let $n$ be a positive integer, and consider all prime knots with $n$ or fewer crossings. The genericity conjecture for hyperbolic knots. The percentage of these knots that are hype…

Let $K$ be an even 3-strand pretzel knot, meaning a pretzel knot with an even parameter in the relevant parity convention. Topological sliceness and ribbonness conjecture. Then $K$…

Let $K$ be a hyperbolic knot in $S^3$, and let $b[?25l\cT_Kb[?25h \in \C[t^{\pm1}]$ be its hyperbolic torsion polynomial. The Seifert genus of $K$ is denoted by $g(K)$. Hyperbolic…

Let $K$ be a Montesinos knot with three branches. Here $M(0; (p,r),(q,s),(2mn\pm 1, 2n^2))$ denotes the Montesinos knot with the displayed parameters. Torisu's conjecture. The knot…

Let $K$ be an achiral knot, and let $C(z)$ denote its Conway polynomial. Kawauchi's conjecture. The polynomial $C(z)$ has the splitting property: there is a polynomial $F(z)$ with…

Let $L$ be a positive integer, let $a_1,\ldots,a_{2L}$ satisfy $a_i\geq 2$, and set $D=a_1+\cdots+a_{2L}$. For the braid word $x_1^{a_1}x_2^{a_2}\dots x_1^{a_{2L-1}}x_2^{a_{2L}}$,…

A Vassiliev invariant is a finite-type invariant of knots, and a quantum Lie group invariant is a knot invariant obtained from quantum-group representations. Bar-Natan's conjecture…

A Vassiliev invariant is a finite-type invariant of knots; a knot is prime if it cannot be expressed as a nontrivial connected sum, and unoriented knots are considered without a ch…

Wheeling conjecture. Wheeling intertwines the two products on Chinese characters; more precisely,

Let ${\frak g}$ be a finite-dimensional Lie algebra with a metric and orthonormal basis $\{{\frak g}_a\}_{a=1}^{\dim{\frak g}}$, and let $R$ be a finite-dimensional representation.…

For each $m$, let ${\cal D}^*_m$ be the dual diagram complex with differential $\partial$, and let $H^1_\partial({\cal D}^*_m)$ denote its first cohomology group. The graph-homolog…

Let an invariant of chord diagrams satisfy the $T4T$ and $TFI$ relations. An invariant is integrated one step when it is lifted to an invariant at the next diagrammatic level. The…