Unsolved Problems

Does every ribbon knot arise as a symmetric union?...

Given $K$ in $S^{3}$, is there an algorithm to detect if $K$ is slice? Ribbon?...

(a) Which knot properties are hereditary under ribbon concordance? Is the property of being alternating hereditary under ribbon concordance? (b) Which...

This problem is concerned with the restriction of the partial ordering $\geq$ coming from ribbon concordance to the concordance class $[K]$ of a knot ...

Suppose that $C$ is a ribbon concordance from a fibered knot $K_{1}$ to a fibered knot $K_{0}$. (a) Does the capped-off monodromy of $K_{1}$ (i.e. ext...

(Hom). If $K_{0}$ and $K_{1}$ are ribbon concordant and $$ \widehat{\mathrm{HFK}}(K_{0}) \cong \widehat{\mathrm{HFK}}(K_{1}), $$ are $K_{0}$ and $K_...

(a) In either the smooth or topological settings, are 0-shake slice knots slice? (b) Does there exist a knot $K$ whose topological 0-shake slice genus...

What concordance information about a knot $K$ is contained in its 0-trace $X_{0}(K)$ and in its 0-surgery $S^{3}_{0}(K)$? Specifically, (a) Suppose $K...

Let $K \subset S^{3}$ be a slice knot. (a) Determine the set $\mathcal{R}(K)$ of ribbon disks bounded by $K$ modulo isotopy. (b) Determine the set $\m...

Is there a knot in $S^{3}$ that is not smoothly slice in $B^{4}$ but is smoothly slice in an integer homology ball? What about a $\mathbb{Z}$/2-homolo...

A knot in $S^{3}$ bounds a topological disk in $B^{4}$ by coning (not necessarily locally flat); this problem asks about topological disks that a knot...

(a) Are all good boundary links topologically slice? Freely topologically slice? (b) A special case of interest: Is the Whitehead double of the Borrom...

Is there a knot type with Legendrian representatives that do not destabilize but have arbitrarily negative Thurston–Bennequin number?...

(a) Let $L \subset (S^{3}, \xi_{std})$ be a transverse link such that the branched double cover $(\Sigma_{2}(L), \xi_{L})$ is Stein fillable. Is $L$ t...

Decomposable Lagrangian cobordisms between Legendrian knots or links in $\mathbb{R}^{3}$ are compositions of certain simple pieces admitting diagramma...

For Legendrian links $\Lambda_{1}, \Lambda_{2} \subset (\mathbb{R}^{3}, \xi_{std})$, write $\Lambda_{1} \preceq \Lambda_{2}$ if there is an exact Lagr...

Given a Legendrian link in the standard contact $\mathbb{R}^{3}$ besides the standard unknot or Hopf link, classify its exact Lagrangian fillings up t...

Determine the smooth knot types that have Legendrian repre- sentatives with orientable exact Lagrangian fillings....

Let $L \subset (S^{3}, \xi_{std})$ be a transverse link with $$ sl_{\Sigma}(L) = -\chi(\Sigma), $$ for some Seifert surface $\Sigma$. Must $L$ be st...

(a) Let $L \subset (S^{3}, \xi_{std})$ be a transverse link with $$ sl_{\Sigma}(L) = -\chi(\Sigma), $$ for some smooth surface $\Sigma \subset B^{4}...

Does a Gordian unknot exist?...

(The equilateral stuck unknots conjecture.). Are there equilat- eral embedded polygons that are unknotted yet cannot be unknotted through polygons pre...

(The 15 pearls conjecture). Is the pearl number of the trefoil equal to 15?...

How does ropelength behave under connected sum of knots? Here are two conjectures, the second a weakening of the first. (a) For any knot or link types...

(a) Find some knot energy on the space of smoothly embedded unknotted circles in $\mathbb{S}^{3}$ for which all unknotted critical points are great ci...

(a) Is there an algorithm to detect the unknot that runs in polynomial time (as a function of the number of crossings in an input diagram)? (b) What i...

How many Reidemeister moves are required to relate two dia- grams of a knot (as a function of their numbers of crossings)?...

Let $D$ be any diagram of the unknot with $n$ crossings. Let $h(D)$ be the smallest number such that some series of Reidemeister moves that transforms...

Are there additional moves that, when added to the three Rei- demeister moves, allow for strict monotonic descent in the crossing number of an unknot ...

Is unknotting number computable? Is there even an algorithm to decide whether a knot has unknotting number one?...

(a) Are all knots trivial? (b) Conjecture: The Bing sling is knotted....

(a) What is a positive knot? (b) Describe a simple set of moves to convert between two positive diagrams of the same knot or link....

Determine the algebraic structure of the concordance group $\mathcal{O}$ of open strings. (a) Is it abelian? (b) Does it contain torsion?...

For a classical knot, does its slice genus as a virtual knot agree with its slice genus as a classical knot?...

Let $K$ be a hyperbolic knot in $S^{3}$ and $\chi(K)$ the space of con- jugacy classes of $\operatorname{PSL}_{2}(\mathbb{C})$ representations of $\pi...

(a) Every connected cubic $($ i.e. trivalent $)$ graph has freeness index at least 2. (b) Every graph has freeness index at least two. (c) There is a ...

Is every fibered link in $S^{3}$ realized as the link of an isolated singular point of a polynomial map $\mathbb{R}^{4} \to \mathbb{R}^{2}$?...

Are there infinitely many congruence arithmetic links in the 3-sphere?...

(a) Fix a long link L. What is the homotopy type of the embedding space of links isotopic to L? (b) Fix a link L in a 3-manifold M. What is the homoto...

(Ivanov conjecture). Let $S$ be an orientable surface of finite type with genus at least three. If $G \leq \operatorname{Mod}(S)$ is a subgroup of fin...

(Congruence subgroup problem). Does every finite-index sub- group of the mapping class group of $S$ contain a congruence subgroup?...

Is the mapping class group of a surface of finite type linear?...

Let $S_{1}$ and $S_{2}$ be orientable surfaces of finite type. Under what conditions do injective maps from (finite-index subgroups of) the mapping cl...

For $g \geq 3$, determine a finite presentation for the Torelli group $\mathcal{I}_{g}$, or show that no finite presentation exists....

Give a classification or enumeration of the finite-index sub- groups of $\operatorname{Mod}(S_{g})$ that are generated by Dehn twists, Dehn multitwist...

Classify the homomorphisms from the braid group $B_{n}$ on $n$ strands to the braid group $B_{m}$ on $m$ strands, where $n, m \in \mathbb{N}$ are arbi...

Fix distinct trivial tangles $\tau_{1}, \tau_{2}$ for which $\tau_{1} \cup\tau_{2}$ is the unknot. Describe the intersection of the associated wicket ...

Is there a nice presentation of the $n$-stranded braid group whose generating set is the set of all positive elementary braid half-twists?...

(a) Is there an efficient algorithm to compute distances in the curve complex of a surface? The input to the algorithm should be the surface and the c...

Find precise estimates for both the extremal and average behav- ior of the simple lifting degree of curves, in terms of combinatorial (e.g., intersect...