Mathematics Problem Archive

(Y. Shinohara [364]) If n = 4 k + 1 with k > 0, is there a knot with determinant n and signature 4?...

(T. Stanford) IsC2 solvable? Does C2 contain a free group?...

(A. Stoimenow) Do positive links of given signature σ have bounded (below) maximal Euler characteristic χ?...

(A. Stoimenow) If a prime knot K can be transformed into its mirror image by one crossing change, is K achiral or (algebraically?) slice?...

(A. Stoimenow) Let n be an odd natural number, different from 1, 9, and 49, such that n is the sum of two squares. Is there a prime alternating achiral...

(V. Turaev) A pair (a finitely generated abelian group H of rank 1, an element ∆( t)∈ Z[H/TorsH] = Z[t±1]) (where t is a generator of H/TorsH ) can be...

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 ...

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

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

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 ....

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...

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...

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...

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

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...

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 ...

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 ...

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....

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”....

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....

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?...

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 $T\mapsto F$....

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 $\rho(f)$?...

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...