Unsolved Problems

Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?...

For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement....

Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales....

Explore homotopy theory's role in Langlands programs....

Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\pi, 1)$ space?...

Can unknots be recognized in polynomial time?...

Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...

Do quantum invariants of knots relate asymptotically to hyperbolic volume?...

Are certain combinations of Pontryagin classes homotopy invariant?...

Is the assembly map in K-theory an isomorphism for all locally compact groups?...

Are Berge knots the only knots in S³ admitting lens space surgeries?...

Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...

If a locally compact group acts faithfully and continuously on a manifold, must it be a Lie group?...

Are certain polynomials in Pontryagin classes homotopy invariants?...

Can unknots be recognized in polynomial time?...

Do quantum invariants of knots determine their hyperbolic volume?...

Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?...

Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?...

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 ...