Unsolved Problems

Is the crossing number additive under connected sum: $c(K_{1}\#K_{2}) = c(K_{1}) + c(K_{2})$?...

(a) Show that if $P$ is a nontrivial satellite operator and $K_{P}$ is a nontrivial satellite of a knot $K$, then $$ c(K_{P}) \geq c(K), $$ where $c...

How does unknotting number behave under connected sum and mutation? (a) Does the connected sum of $n$ nontrivial knots have unknotting number at least...

Let $P$ be a nontrivial satellite pattern with winding number $w(P) \neq 0$. Then for any nontrivial knot $K$ and its satellite $K_{P}$ , one has $$ ...

Is there a relationship between genus and unknotting number for specific classes of knots? Here are two instances of classes of knots for which there ...

Suppose that $V_{1}$ and $V_{2}$ are $S$–equivalent Seifert forms. Does there exist a fixed knot $K$ bounding Seifert surfaces $F_{1}$ and $F_{2}$ for...

Show that the sequence of absolute values of the coefficients of the Alexander polynomial of a link are: (a) concave $($ Fox’s trapezoidal conjecture ...

Which multi-variable Laurent polynomials arise as the multi- variable Alexander polynomial of a link in the 3-sphere or, more generally, a ho- mology ...

If Dehn surgery on a knot $K$ gives a lens space, then $K$ is a Berge knot....

(Generalized Property R Conjecture). Let $L \subset S^{3}$ be an $n$- component link such that 0-framed Dehn surgery on $L$ results in $\#^{n}(S^{1} \...

(Cabling conjecture). Let $K \subset S^{3}$ be a knot and $r \in \mathbb{Q}$. If $r$-framed Dehn surgery on $K$ is not prime, then $K$ is a nontrivial...

This problem presents several variations on the Cosmetic Surgery Conjecture, discussed in turn below. (a) $($ Cosmetic Surgery Conjecture $)$ Two surg...

Let $K$ be a null-homotopic knot in a 3-manifold $Y$ , and let $Y_{0}(K)$ be the manifold obtained by 0-surgery on $K$. (a) Conjecture: Let $F$ be a S...

For which nonzero $r \in \mathbb{Q}$ is it true that for every nontrivial knot $K \subset S^{3}$ there is a homomorphism $$ \pi_{1}(S^{3}_{r}(K)) \to...

(a) Are there integral homology spheres with arbitrarily large (integral) Dehn surgery number? Are there irreducible examples? Does the connected sum ...

(a) Given a knot $K \subset S^{3}$ determine all knots $K' \subset S^{3}$ for which the branched double covers of $S^{3}$ along $K$ and $K'$ are homeo...

Can an alternating link and a non-alternating link have home- omorphic branched double covers?...

(a) $($ Meridional Rank Conjecture $)$ Is the meridional $\operatorname{rank} \mu(L)$ of every link $L$ equal to its bridge number $b(L)$? (b) Given t...

Let $Y = Y_{1}\#Y_{2}$ be a connected sum of 3-manifolds with $Y_{i} \neq$ $S^{3}$, for $i = 1, 2$. Let $\Phi: Y \to Y$ be a Dehn twist around the con...

(a) Are there any null-homologous Floer minimal knots with irreducible com- plements other than the Borromean knots $B_{g}, g \geq 0$, in any 3-manifo...

(a) For a given positive integer $g$, are there only finitely many L-space knots of genus $g$? A related but more general question is: (b) Question (H...

If $K$ is a hyperbolic $L$-space knot, show that its branched cover $\Sigma_{2}(K)$ is not an $L$-space....

Let $K$ be a cubic graph embedded in the plane, and let $\mathrm{Tait}(K)$ be the number of Tait colorings of $K$. (a) Is $\dim J^{7}(K) = \mathrm{Tai...

(Jones Unknot Detection). (a) Is there a nontrivial knot with the same Jones polynomial as the unknot? (b) Does there exist a nontrivial knot whose c...

(a) Conjecture: The noncommutative $A$-ideal of a knot $K$ is exactly the an- nihilator of the colored Jones polynomial $($ the infinite dimensional v...

(Jones Slope Conjecture). For a knot $K$, the Jones slopes $js(K)$ are the set of cluster points in $$ \left\{\frac{4}{n^{2}}\deg_{+}\bigl(J_{n}(K;q)...

(Kashaev–Murakami–Murakami Volume Conjecture). For a link $L \subset S^{3}$, $$ \frac{1}{2\pi}\operatorname{Vol}_{\mathrm{hyp}}(S^{3}-L) =\lim_{n\to\...

Let $L$ be a link in the thickened annulus $S^{1} \times I \times I$. (a) Wrapping conjecture: $w(L)$ is equal to the maximal nonzero annular de- gree...

Is the first inequality below true? If so, is the second? (a) $(\operatorname{Vol}$-Det Conjecture $)$ For any alternating hyperbolic knot $K \subset ...

Does the Khovanov homology of every nontrivial knot contain 2-torsion?...

(a) Compute the Khovanov homology for all torus knots $T(m, n)$. (b) Compute the Khovanov–Rozansky $\mathfrak{g}\mathfrak{l}(N)$ homology for all toru...

(a) Recover the Jones polynomial of links $L \subset \mathbb{R}^{3}$ by counting solutions to the Kapustin–Witten equations on $\mathbb{R}^{3} \times ...

Describe topological necessary or sufficient conditions for a link to have KR-parity. For example: (a) Are all links with KR-parity positive? Quasipos...

(a) Khovanov and Rozansky [KR08b] used braid presentations to define a triply graded link homology theory whose Euler characteristic is the HOM- FLYPT...

(a) Is symplectic Khovanov homology isomorphic to Khovanov homology, over $\mathbb{Z}$? (b) Give a construction of odd symplectic Khovanov homology $\...

Categorify the ($\mathfrak{s}\mathfrak{l}(2), \mathfrak{s}\mathfrak{l}(N)$, HOMFLYPT) skein algebras for surfaces....

(a) For every link $L \subset \mathbb{R}^{3}$, every simple Lie algebra $\mathfrak{g}$, and every coloring of the components of $L$ with irreducible r...

What is the structure of the smooth knot concordance group? (a) Is there a torsion element of the smooth concordance group $\mathcal{C}$ having order ...

(a) Do the algebraic knots freely generate a subgroup of the smooth concor- dance group $\mathcal{C}$? (b) Do the algebraic knots freely generate a su...

A satellite operator $P \subset S^{1} \times D^{2}$ induces an operation $P$ on the concordance group $\mathcal{C}$ [Gor75]. (a) Let $P$ be a winding ...

This problem is concerned with the stable 4-genus $g_{s}(K)$ of a knot $K$, defined below. (a) Is there a knot $K$ such that $g_{s}(K) \in \mathbb{Q}\...

Do there exist algebraically concordant Seifert forms $V_{1}$ and $V_{2}$ for which there do not exist concordant knots $K_{1}$ and $K_{2}$ with Seife...

Does knot Floer homology give a categorification of the Fox– Milnor condition?...

(a) If $K \in \mathcal{F}_{n}$ for all $n$, is $K$ topologically slice? (b) If $K \in \mathcal{T}_{n}$ for all $n$, is $K$ smoothly slice?...

(a) For arbitrary $n \geq 2.5$ and $g > 1$, does there exist a knot in $\mathcal{F}_{n}$ with topological slice genus at least $g$? (b) For arbitrary ...

(a) Determine the topological slice genera of torus knots. In particular, does the topological slice genus of a torus knot equal half the absolute val...

Given a smooth knot $K \subset S^{3}$, determine its nonorientable 4- genus $\gamma_{4}$, i.e. the minimal first Betti number for all compact nonorien...

(a) Suppose $K$ and $K\#J$ are (smoothly) doubly slice knots. Must $J$ be a (smoothly) doubly slice knot? (b) Does there exist a knot that is smoothly...

(a) What is the structure of the equivariant concordance groups? (b) Is the strongly negative amphichiral concordance group abelian? (c) For any type ...

(a) Is every slice knot a ribbon knot? (b) Is every slice link ribbon? (c) Suppose $K$ is a knot with smooth four-genus $g_{4}(K) = g$. Does $K$ bound...