1,852 problems

Let $K$ be a nontrivial knot, and let $K(r)$ denote the result of $r$-Dehn surgery on $K$ for $r\in\mathbb{Q}$. Property P conjecture. The manifold $K(r)$ is never simply connected…

Cosmetic surgery conjecture. The conjecture predicts that non-trivial knots admit no truly cosmetic surgeries.

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 $T_{p,q}$ be a torus knot with parameters $p$ and $q$. Milnor's conjecture. The smooth 4-genus of $T_{p,q}$ is … The conjecture is a theorem, proved by Kronheimer and Mrowka, a…

Let $K$ be a knot, and let $V_K(x)$ denote its Jones polynomial in the normalization used here. Jones polynomial unknot-detection conjecture. … The conjecture asks whether the Jone…

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 $K\subset S^3$ be a hyperbolic knot, let $g(K)$ denote its genus, and let $\mathcal T_K(t)\in\mathbb C[t^{\pm1}]$ be the normalized hyperbolic torsion polynomial associated wit…

Morse-Novikov additivity conjecture. One should have

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…

Knots in $S^3$ are related by ribbon concordance when there is a ribbon concordance between them. Gordon's conjecture. Ribbon concordance defines a partial order on knots in $S^3$.…

Let $K$ be a knot in $S^3$ and let $r,s e r$ be slopes in $mathbb{Q}cup{infty}$. Write $S^3r(K)$ for the manifold obtained by Dehn surgery on $K$ with slope $r$; two surgeries are…

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

Goda–Teragaito lens space surgery conjecture. Then $K$ is a fibered knot and

Let $K$ be a knot. The unreduced Khovanov homology $Kh(K;\mathbb{Z})$ is the Khovanov homology of $K$ with integer coefficients, and $2$-torsion means an element of order $2$. Shum…

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…

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

Let $K\subset S^3$ be a knot. It is slice if it bounds a smooth properly embedded disk in the standard $4$-ball $B^4$, and it is ribbon if it bounds a disk that can be isotoped so…

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…

Let $K$ be a hyperbolic L-space knot, let $g$ be its Seifert genus, and let $n$ be its braid index. Let $Delta_K(t)$ denote its Alexander polynomial. The hyperbolic L-space knot Al…

Determinant-square conjecture. In general, the maximum determinant of a prime symmetric ribbon knot $K$ with $r_s(K)=n$ is the square of the maximum determinant of a prime knot $J$…

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…

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…

Let $K$ be a knot, let $J_{K,n}(q)\in\mathbb{Z}[q^{\pm1}]$ be its $n$-th colored Jones polynomial, and define … Let $\triangle_K$ denote the Alexander polynomial of $K$. Melvin–Mor…

Let $K$ be a knot. Its colored Jones polynomial is the sequence of quantum knot invariants indexed by colors, and its A-polynomial is the algebraic curve encoding the boundary char…

Residual torsion-free nilpotence conjecture. The commutator subgroup of $\pi_1(M_J)$ is residually torsion-free nilpotent if and only if $\Delta_J$ is nontrivial.