Mathematics Problem Archive

Let $VKT/Z$ denote virtual knot theory modulo $Z$-equivalence, as defined in the section above on virtual knot theory. Recall that two virtual diagram...

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...

Biquandles: The first two problems are old chestnuts. Give a descriptive representation of the free biquandle. Give a topological explanation of the...

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 ...

In we show how, by translating between knots and planar graphs (the checkerboard and medial constructions) one can associate a signed graph to a class...

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

The Kauffman bracket polynomial of a virtual diagram can have the leading term, i.e., the term having the highest possible degree, equal to zero. Ass...

We still do not know whether the free knot whose Gauss diagram is a heptagon (i.e. consists of 7 chords each of which is linked with precisely two adj...

Given a chord diagram $D$ we can construct the following formal chain complex. Formally speaking, this complex will look like a simplicial complex in ...

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....

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 $T\mapsto F$....