Unsolved Problems

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...

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....