947 problems

For every modular knot $K$, construct explicitly a weight-$2$ modular integral $F_K$ for $\mathrm{SL}_2(\mathbb{Z})$ such that, for every modular knot $L$, the homogenized cycle in…

For every annular link $L$ in a solid torus, the Kauffman bracket skein element $\langle L\rangle$ has a nonzero homogeneous component in annular degree $\operatorname{wrap}(L)$; e…

For every knot $K$, there exists a minimal-crossing diagram $D$ of $K$ and a crossing of $D$ such that changing that crossing produces a knot $K'$ satisfying $u(K')=u(K)-1$, where…

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For every smooth knot type $K$ that is Legendrian simple—meaning that any two Legendrian representatives $L_1,L_2$ of $K$ with $\operatorname{tb}(L_1)=\operatorname{tb}(L_2)$ and…

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

A 4-move is the local operation on a link diagram that replaces four consecutive half-twists between two strands by the opposite four half-twists. Nakanishi's 4-move conjecture. Ev…

Jones's braid-index uniqueness conjecture. If

Let ${\mathcal K}$ be a tame knot class, let $\operatorname{bra}{\mathcal K}$ denote its braid index, and let $\operatorname{bri}{\mathcal K}$ denote its bridge index. An $a$-times…

Let ${\mathcal D}$ be an alternating knot diagram with no nugatory crossings. The determinant of ${\mathcal D}$ is the order of its first homology group of the double branched cove…

Let $K$ be an alternating knot, and write its Alexander polynomial as $\Delta_K(t)=\sum_i a_i t^i$. A sequence is trapezoidal if the absolute values of its terms increase, possibly…

Let $K$ be a hyperbolic knot in $S^3$, let $E_K=S^3\setminus\nu(K)$ be its exterior, and let $x(K)$ denote its Thurston norm, equivalently … Let $\rho^{\mathrm{geom}}$ be a lift of…

Garoufalidis's Slope Conjecture. For any knot $K$ in $S^3$, every Jones slope is a boundary slope:

Morse-Novikov additivity conjecture. One should have

Let $K\subset S^3$ be a knot, and let $c$ be a crossing of $K$. A crossing is nugatory if its crossing circle bounds a disk in the complement of $K$; a generalized crossing change…

Hoste's conjecture. Then $\Re(t) > -1$. This conjecture concerns the location of zeros of Alexander polynomials for alternating knots. The source recalls it while analyzing zeros i…

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.

Montesinos–Nakanishi 3-move conjecture. Every link can be reduced to a trivial link by 3-moves.

Let $K$ be a nontrivial knot in $S^3$. For surgery slopes $r$ and $s$, the manifolds $S^3_r(K)$ and $S^3_s(K)$ are purely cosmetic when they are homeomorphic as oriented manifolds.…

Let $S^{3}\setminus K$ be the complement of a hyperbolic knot $K$ in $S^{3}$. A closed embedded totally geodesic surface is a closed totally geodesic surface embedded in this hyper…

Let $K\subset S^3=\partial D^4$ be a knot. It is slice if it bounds a smoothly, properly embedded disk in $D^4$; it is ribbon if it bounds a disk whose only singularities are ribbo…

Jones polynomial conjecture. The Jones polynomial detects causality between any two events in $X$.

Let $J$ be a knot with bridge spectrum … and let $K={\mathsf {cable}}(T_{m,n},J)$ be a cable of $J$. The bridge-spectrum cabling conjecture. … The conjecture expresses the observed…

A knot $K$ is chiral if it is not equivalent to its mirror image, and $T(2,n)$ denotes the $(2,n)$ torus knot. Chirally cosmetic surgery conjecture. If $K$ is a chiral knot that is…