Mathematics Problem Archive

(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...

(a) What is the maximum number of systoles on a closed, hyperbolic surface of genus $g$? (b) What is the maximum cardinality of a set of pairwise non-...

Let $S$ be a surface, and let $\Gamma_{1}, \Gamma_{2}$ be isotopy classes of embedded graphs in $S$. Determine when $\Gamma_{1}$ and $\Gamma_{2}$ are ...

(a) Does every Jordan curve in the Euclidean plane contain the vertices of a square? (b) Does every Jordan curve in the Euclidean plane contain the ve...

Is the genus $g$ Goeritz group $\mathcal{G}_{g}$ finitely generated when $g \geq 4$? If so, find a set of generators....

If two hyperbolic surfaces have the same unmarked simple length spectra (i.e., the same multiset of lengths that correspond to simple closed curves), ...

Suppose that $c$ is a geodesic current on a hyperbolic surface, and suppose that, on the space of hyperbolic metrics on the surface, $c$ has the same ...

What is the best lower bound on the volume of a fibered hyper- bolic 3-manifold one can give in terms of the translation length of the monodromy with ...

Which mapping classes give rise to arithmetic hyperbolic 3- manifolds as their mapping tori?...

Can one detect holomorphicity from a monodromy factoriza- tion of a Lefschetz pencil, fibration, or a surface bundle over a surface? What are the spec...

What is the minimum number, $m_{g,b}$, of right-handed Dehn twists along essential curves into which the boundary multi-twist, $\Delta:= T_{\delta_{1}...

Does every Lefschetz fibration over the 2–sphere admit a sec- tion?...

Does there exist a surface bundle over a surface that admits a complete hyperbolic metric, or, more generally, a complete metric of variable negative ...

Does there exist a complex surface $X$ that admits three or more non-isomorphic structures as a surface bundle over a surface?...