Mathematics Problem Archive

Long Knots and Long Flat Knots: Enlarge the long knot invariant structure proposed in . Can one get new classical knot invariants from the approach in...

Virtual braids: Is there a birack such that its action on virtual braids is faithful? Is the invariant of virtual braids in , see also , faithful? T...

Khovanov Homology: Construct a generalization of the Khovanov complex for the case of virtual knots that will work for arbitrary virtual diagrams. Inv...

Brauer algebra: The appropriate domain for the virtual recoupling theory is to place the Jones–Wenzl projectors in the Brauer algebra. That is, when w...

Virtual Alternating Knots: Define and classify alternating virtual knots. Find an analogue of the Tait flyping conjecture and prove it. Compare . Cl...

Crossing Number: One of the most important problems in knot theory is the problem of finding the minimal crossing number for a given knot. It can be e...

Crossing number problems: For each virtual link $L$, there are three crossing numbers: the minimal number $C$ of classical crossings, the minimal numb...

Vassiliev Invariants: Understand the connection between virtual knot polynomials and the Vassiliev knot invariants of virtual knots (in Kauffman's sen...

The Rack Space: The rack space was invented by Fenn, Rourke and Sanderson . The homology of the rack space has been considered by the above authors an...

Rotational virtual knot theory introduced in is virtual knot theory without the first virtual move (thus one does not allow the addition or deletion o...

The arrow polynomial generalizes to a more powerful invariant of virtual knots by defining a version of it for knots in specific thickened surfaces. T...

Study concordance and cobordism invariants of virtual knots. In particular, solve the question of virtual knots up to pass-equivalence (taking a direc...

Khovanov homology for virtual knots and links works directly with mod-$2$ coefficients. With mod-2 coefficients there are no technical difficulties as...

In we have studied a mod-2 categorification of the arrow polynomial and discovered many baffling examples of pairs of virtual knots that are discrimin...

Make a systematic study of Vassiliev invariants for virtual knots and links. There is ongoing work on this problem....

Generalize virtual knot theory to virtual 2-spheres in 4-space....

Find a way to effectively compute the Kauffman–Radford–Hennings (KRH) invariants for three-manifolds via right integrals on finite dimensional Hopf al...

Create a combinatorial homotopy theory for Khovanov homology so that the Khovanov homology of a knot or link is equivalent to the homotopy type of an ...

In we give a formula for the Kauffman polynomial, due to Jaeger, that expresses this invariant as a state sum over oriented, partially smoothed links ...

It has been pointed out that the chain complex for Knot Floer Homology (categorifying the Alexander–Conway polynomial) is generated by the states desc...

Which quantum invariants extend to virtual knots themselves without restrictions? Certainly, there are such ones, e.g., the Kauffman bracket and the J...

In his wonderful paper , which firstly had the title “Knot Floer Homotopy”, Sarkar constructs a cell complex. The homology of this complex coincides w...

In it was showed how one could determine non-invertibility of long virtual knots and non-commutativity of long virtual knots. This method used two typ...

It is well known that flat virtual knots are easily algorithmically recognizable, see . The absence of a geometric approach to the definition of free ...

The functorial mapping was constructed, by means of it we constructed the map from the set of virtual knots to the set of virtual knots with orientabl...

Is there any algorithm for recognition whether two graph-links are equivalent or not? Our conjecture is “no”. The idea behind that is that graph-links...

Problems on Free Knot Cobordism: The methods used for proving the fact that the invariant $L$ gives an obstruction to the sliceness are not immediatel...

Is string topology sensitive to smooth structures on 4-manifolds?...

What are the obstructions to the realization of these chains of formal changes by paths in the parameter space ${\mathbb{R}}^{k}$?...

Are these geometric obstructions sufficient to solve the above problem?...

Is it true that any hypersurface from the space $P(d;N)$ can be connected with a trivial one by a generic path in this space in such a way that it exp...

Do non-singular real plane projective curves of an odd degree consisting of a single connected component form a connected set (i.e., are they rigid is...

Is it true that any component of the complement of the discriminant variety of a versal deformation contains a Morsification, whose all $\mu(f)$ criti...

Let $N$ be the closed $3$-manifold obtained by gluing two copies of the figure-eight knot complement along their torus boundaries by a homeomorphism. ...

Let $N$ be a closed $3$-manifold whose JSJ decomposition contains only hyperbolic pieces. Does $N$ admit a convex projective structure?...

Geometric transitions are continuous paths of geometries that abruptly change type in the limit. Understand all transitions between the eight Thurston...

Do cusped finite-volume hyperbolic $3$-manifolds have closed quasi-Fuchsian surfaces that cubulate except for the cusps? Equivalently in the stated ge...

Which Kleinian groups admit closed quasi-Fuchsian surface subgroups?...

Let $M$ be a $3$-manifold, let $\phi:\pi_1M\to\mathbb{Z}$ be dual to $(\Sigma,\partial\Sigma)\subset(M,\partial M)$, and let $N$ be obtained by cuttin...

Is there a practical algorithm to test whether a pair of $3$-manifolds are homeomorphic?...

The CAT(0) cubical dimension of a group $G$ is the least dimension of a CAT(0) cubical space on which $G$ acts geometrically. Is there a sequence of c...

Let $\Gamma$ be a discrete group acting properly discontinuously by affine transformations on $\mathbb{R}^5$. Is $\Gamma$ virtually an extension of a ...

Classify all properly discontinuous affine actions of a given closed surface group on $\mathbb{R}^6$....

For a given right-angled Coxeter group, what is the least $n$ for which it admits a proper affine action on $\mathbb{R}^n$?...

Characterize the Thurston norm polytopes of finite-volume hyperbolic $3$-manifolds....

The renormalized volume of quasi-Fuchsian groups gives a function $\rho:\mathcal{T}(S)\times\mathcal{T}(S)\to\mathbb{R}$. Is $\rho$ a metric on the Te...

Build a combinatorial model for hyperbolic $3$-manifolds with explicit bilipschitz constants....

Let $\Gamma$ be a Kleinian group of finite covolume, and let $\mathcal{C}(\Gamma)$ be the set of isomorphism classes of its finite quotient groups. Do...

For $\Gamma=F_r$ with $r\geq2$, does $\mathcal{C}(F_r)$ determine $F_r$ up to isomorphism?...

Let $\Gamma$ be a Kleinian group of finite covolume. Does its set $\mathcal{C}(\Gamma)$ of finite quotient isomorphism classes determine $\Gamma$ amon...