Mathematics Problem Archive

The Flat Hierarchy: The flat hierarchy is constructed for any ordinal $\alpha$. We label flat crossings with members of this ordinal. In a flat third ...

Virtuals and the Theory of Doodles: Compare flat theories of virtual knots with theories of doodles. A doodle is represented by a flat diagram in the ...

Virtual Three Manifolds: There is a theory of virtual $3$–manifolds constructed as formal equivalence classes of virtual diagrams modulo generalized K...

Welded Knots: We would like to understand welded knots . It is well known that if we admit forbidden moves to the virtual link diagrams, each virtual ...

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 Biquandle: Construct presentations of the virtual biquandle with the a linear (non-commutative) representation at classical crossings and some...

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...

The Fundamental Biquandle: Does the fundamental biquandle, see classify virtual links up to mirror images? (We know that the biquandle has the same va...

Virtualization and Unit Jones Polynomial: Suppose the knot $K$ is classical and not trivial. Suppose that ${\tilde K}$ (obtained from $K$ by virtualiz...

Virtual Quandle Homology: Study virtual quandle homology in analogy to quandle homology ....

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...

Wild Virtuals: Create the category of “wild virtual knots” and establish its axiomatics. In particular, one needs a theorem that states when a wild eq...

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

Embeddings of Surfaces: Given a non-trivial virtual knot $K$. Prove that there exists a minimal realization of $K$ in $N=S_{g}\times I$ and an unknott...

Non-Commutativity and Long Knots: It is known that any classical long knot commutes with any long knot. This is definitely not the case in the virtual...

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...

Find new geometric/topological interpretations for the Jones polynomial and for Khovanov homology....

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....