Mathematics Problem Archive

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

One can consider braids with even numbers of strands. Markov's moves change the parity of the number of strands. Can one reformulate Markov's theorem ...

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

A complete invariant is used for proving that one theory is a part of another theory. For example, the fact that the set of classical knots is a part ...

To prove that the invariant $\mathcal{F}$ constructed by V. O. Manturov for virtual braids, is complete for virtual braids with more than two strands....

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...

Turaev constructed the map from the set of long flat knots to the set of long virtual knots. Can one construct any map from the set of long free knots...

The problem about cobordisms in sections: Let us have a free knot and its cobordism. Construct a parity on the given free knot, which is defined by us...

Prove or disprove the conjecture about the non-uniqueness of minimal representative of a free link, i.e. there exists a free link having several minim...

If $X$ is the free rack, is $\Gamma X$ a cat(0) space? A positive answer would imply that all its higher homotopy groups are trivial....

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...

Can one construct a projection from the set of graph-links to the set of realizable graph-links?...

Construct a parity on graph-links by using Bouchet's criterion about the realizability of a graph. Try to find a parity which is responsible for the c...

Construct generalizations of the Frobenius extension and the Rasmussen for “rigid” graph-links with orientable atoms....

Is it true that two equivalent realizable graph-links are equivalent in the class of realizable graph-links? If it is not true, then construct an exam...

Construct a group for graph-links which is analogous to the group from ....

Construct “graph-braids”....

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?...

Present explicit obstructions to the Lyashko–Looijenga covering in terms of braid groups. Which braids cannot be lifted to the space ${\mathbb{C}}^{\m...

Let a non-simple singularity $f$ be given and the Dynkin diagram of it be defined by an easily distinguished system of paths connecting 0 with critica...

Are there more refined restrictions to the collision of critical values? Is it true that for any two vanishing cycles, whose intersection number is eq...

Give more general lower bounds of the dimension of $\mu=const$ strata in terms of intersection forms of vanishing cycles....

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

Is there any convenient topological characteristic of the function $f_{-\varepsilon}$ which allows to predict these indices?...

What is the minimal number of open sets $U_{i}$ covering ${\mathbb{R}}^{6}$ such that for any $U_{i}$ there is a continuous map $\varphi_{i}:U_{i}\to{...

The same questions concerning the approximate solutions. That is, for any $i$ and any $(a,b)\in U_{i}$, the value $\varphi_{i}(a,b)$ should be not nec...

Is the complement of the essential ramification set in ${\mathbb{R}}^{d}$ a $K(\pi,1)$-space?...

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...

A version of the previous problem, in which the complexity measure is not purely topological: namely, it is the lowest number of critical points of Mo...

Give an upper bound for the function $T\mapsto F$....

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 real Morsification of $f$ can be connected with one of complexity $\rho(f)$ by a generic path in the base of a versal deformation,...

What can be said about the number $\rho(f)$?...

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...

Give a similar universal estimate of the radius of convergence for multidimensional Newton’s method of [Shub and Smale1993]....

If $M^n$ is a closed manifold with a strictly convex projective structure, is it cubulated?...

Suppose a hyperbolic $3$-manifold $M$ admits a CR structure, not necessarily a spherical one. Can the deformation theory of convex real projective str...

If $M$ is a closed hyperbolic $3$-manifold, is every projective structure on $M$ convex?...

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...

Does there exist a nonzero polynomial on $U\otimes V\otimes W$ invariant under $SL(U)\times SL(V)\times SL(W)$ when $\dim U=\dim V=4$ and $\dim W=8$?...

Let $\beta$ be a nondegenerate bilinear form on a finite-dimensional real vector space $V$, and let $G=\operatorname{Isom}(\beta)\subset GL(V)$. Which...

Let $M$ be a $3$-manifold whose fundamental group acts on a simplicial tree without global fixed points. In the covering space associated to an edge s...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with only double curves of intersection?...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with the 1-line property: in the universal cover, every pair of p...