We construct a series of combinatorial quandle-like knot invariants. We color regions of a knot diagram rather than lines and assign a weight to each coloring. Sets of these weights are the invariants we construct (colorings and weights depend on several parameters).

Using these invariants, we prove that left and right trefoils are not isotopic using this invariant (in a particular case).

A link map is a map of spheres into another sphere with pairwise disjoint images, and a link homotopy is a homotopy through link maps. In this talk I will discuss the problem of classifying, up to link homotopy, two-component link maps of two-spheres in the four-sphere. This setting is particularly interesting because, as usual, four-dimensional topology presents unique difficulties. After giving a brief history of the subject, I will describe how invariants of four-dimensional link homotopy arise as obstructions to equipping a link map with Whitney disks, which are the devices for performing the so-called Whitney trick. I will then discuss a result that says an invariant due to Kirk detects a certain nice variety of such Whitney disks.

The paper addresses the enumeration problem for k-tangles. We introduce the notion of a cascade diagram of a k-tangle projection and suggest an effective enumeration algorithm for projections based on the cascade representation. Tangle projections and alternating tangles with up to 12 crossings are tabulated. We also provide pictures of alternating k-tangles with at most five crossings.

A sequence of FF-polynomials \{FnK(t,ℓ)\}∞n=1\{FKn(t,ℓ)\}n=1∞ of virtual knots KK was defined by Kaur \emph{et al.} in 2018. These polynomials have been expressed in terms of index value of crossing and nn-writhe of KK. By the construction, FF-polynomials are generalizations of Kauffman's Affine Index Polynomial, and are invariants of virtual knot KK. We present values of FF-polynomials of oriented virtual knots having at most four classical crossings in a diagram.

We construct an infinite series of invariants of Fiedler type (i.e. composed of oriented arrow diagrams arranged by elements of H1(M3)) for multicomponent links in M^3 = M^2 × R^1, M^2 orientable with П_1(M^2) ≠ {1}.

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in S^3. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using v-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

A loop S^1 → ℝ^n is holonomic if it is the (n - 1)-jet extension of a function S^1 → ℝ^1. We prove that for n = 3 any tame link in ℝ^n is isotopy equivalent to a holonomic one; for n > 3 the space of holonomic links is holotopy equivalent to the space of all differentiable links.

We modify the definition of the Khovanov complex for oriented links in a thickening of an oriented surface to obtain a triply graded homological link invariant with a new homotopical grading.

The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and arbitrary bipartite graphs. It turns out to be isomorphic to the algebra of differential operators from [18] which serves a model for open-closed string theory. The operator associated with the Young diagram of the transposition of two elements coincides with the cut-and-join operator which gives rise to relations for the classical Hurwitz numbers. We prove that the operators corresponding to arbitrary Young diagrams and bipartite graphs also give rise to relations for the Hurwitz numbers.

An infinite family of invariants of multicomponent links in 3-manifolds is introduced and used to prove the non-splitting and non-equivalence of textile structures.

A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested.

The construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.

An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can be reduced to the case when the extreme coefficient function is expressible in terms of chord diagrams (previous authors consider only d-diagrams which correspond to the classical case [9]). We find the maximal absolute values for generic chord diagrams, thus, for generic virtual knots. Also we consider the "next" coefficient of the Jones–Kauffman polynomial in terms of framed chord diagrams and find its maximal value for a given number of chords. These two functions on chord diagrams are of their own interest because there are related to the Vassiliev invariants of classical knots and J-invariants of planar curves, as mentioned in [10].

We fill a gap in the proof that the proposed link homotopy invariant w of Li is well defined. It is also shown that if the homotopy invariant τ of Schneiderman–Teichner is to be adapted to a link homotopy invariant of link maps, the result coincides with w.

Typical examples of textile structures are separated by finite type invariants of knots in non-trivial (in particular, non-orientable) manifolds. A new series of such invariants is described.

To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of L-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of L-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

We study perturbations of the maximal stable state in a sandpile model on the set of faces of the heptagonal tiling on the hyperbolic plane. An explicit description for relaxations of such states is given.

We consider a braid β which acts on a punctured plane. Then we construct a local

system on this plane and find a homology cycle D in its symmetric power, such that D · β(D) coincides with the Alexander polynomial of the plait closure of β.

We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labeled graphs is obtained.