A characterization of alternating links in thickened surfaces

We use an extension of Gordon–Litherland pairing to thickened surfaces to give a topological characterization of alternating links in thickened surfaces. If $\Sigma$ is a closed oriented surface and $F$ is a compact unoriented surface in $\Sigma \times I$ , then the Gordon–Litherland pairing defines a symmetric bilinear pairing on the first homology of $F$ . A compact surface in $\Sigma \times I$ is called definite if its Gordon–Litherland pairing is a definite form. We prove that a link $L$ in a thickened surface is non-split, alternating, and of minimal genus if and only if it bounds two definite surfaces of opposite sign.

Khovanov–Lipshitz–Sarkar homotopy type for links in thickened higher genus surfaces

We discuss links in thickened surfaces. We define the Khovanov–Lipshitz–Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. A surface means a closed oriented surface unless otherwise stated. Of course, a surface may or may not be the sphere. A thickened surface means a product manifold of a surface and the interval. A link in a thickened surface (respectively, a 3-manifold) means a submanifold of a thickened surface (respectively, a 3-manifold) which is diffeomorphic to a disjoint collection of circles. Our Khovanov–Lipshitz–Sarkar stable homotopy type and our Steenrod square of links in thickened surfaces with genus [Formula: see text] are stronger than the homotopical Khovanov homology of links in thickened surfaces with genus [Formula: see text]. It is the first meaningful Khovanov–Lipshitz–Sarkar stable homotopy type of links in 3-manifolds other than the 3-sphere. We point out that our theory has a different feature in the torus case.

Complexity of virtual multistrings

A virtual[Formula: see text]-string [Formula: see text] consists of a closed, oriented surface [Formula: see text] and a collection [Formula: see text] of [Formula: see text] oriented, closed curves immersed in [Formula: see text]. We consider virtual [Formula: see text]-strings up to virtual homotopy, i.e. stabilizations, destabilizations, stable homeomorphism, and homotopy. Recently, Cahn proved that any virtual 1-string can be virtually homotoped to a minimally filling and crossing-minimal representative by monotonically decreasing both genus and the number of self-intersections. We generalize her result to the case of non-parallel virtual [Formula: see text]-strings. Cahn also proved that any two crossing-irreducible representatives of a virtual 1-string are related by isotopy, Type 3 moves, stabilizations, destabilizations, and stable homeomorphism. Kadokami claimed that this held for virtual [Formula: see text]-strings in general, but Gibson found a counterexample for 5-strings. We show that Kadokami’s statement holds for non-parallel [Formula: see text]-strings and exhibit a counterexample for general virtual 3-strings.

Normal forms for strong magnetic systems on surfaces: trapping regions and rigidity of Zoll systems

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings.

CELLULAR AUTOMATON WITH PERCOLATION AS A DYNAMIC SYSTEM: ENTROPY APPROACH

A model of information confrontation based on a two-dimensional percolation-cellular automaton on a closed oriented surface is constructed and implemented programmatically. Numerical experiments were performed. A theorem on the finiteness of a completely positive topological entropy of a given cellular automaton is formulated and proved. As an applied application of the constructed automaton, a retrospective forecast of the results of the Russian Presidential election in 2018 was made both in Russia as a whole and in two regions of Russia. The General logic of using the automaton was as follows. At the preparatory stage, a model of the region was built, which is a field of a cellular automaton. In the case of regions, anamorphic mapping was used, in which each territorial-administrative unit is represented by a certain number of cells that occupy a connected area of the field of the automaton; the areas of these areas are proportional to the number of voters, and if possible, geographical neighbors are preserved. The color of the cell corresponds to a certain political position; for example, if a certain city has 60% support for the President, then this percentage of cells in that city is red. As initial data for the calculation, the results of sociological surveys on support for the current President conducted 6-12 months before the election were taken; the dynamics of the system is modeled using a cellular automaton; for the formed stationary solution, the shares of cells corresponding to the number of supporters and opponents of the President, as well as non-appearance, are calculated; these shares are taken as a forecast of election results. Those constructed in this way showed significantly lower accuracy than forecasts made using standard sociological methods in the last days before the election. However, they can be used for early forecasting. The reason for this difference is that the opinions of voters immediately prior to elections are determined by campaign, and long before the election - the prevalence and embeddedness of value orientations, which is incorporated in the model.

Filling of closed surfaces

Let [Formula: see text] denote a closed oriented surface of genus [Formula: see text]. A set of simple closed curves is called a filling of [Formula: see text] if its complement is a disjoint union of discs. The mapping class group [Formula: see text] of genus [Formula: see text] acts on the set of fillings of [Formula: see text]. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of [Formula: see text] are in the same [Formula: see text]-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of [Formula: see text] whose complement is a single disc (i.e. a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of [Formula: see text]. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of [Formula: see text] is two. Finally, given positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we construct a filling pair of [Formula: see text] such that the complement is a union of [Formula: see text] topological discs.

Doodles on surfaces

Doodles were introduced in [R. Fenn and P. Taylor, Introducing doodles, in Topology of Low-Dimensional Manifolds, Lecture Notes in Mathematics, Vol. 722 (Springer, Berlin, 1979), pp. 37–43] but were restricted to embedded circles in the [Formula: see text]-sphere. Khovanov [M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997) 2297–2315] extended the idea to immersed circles in the [Formula: see text]-sphere. In this paper, we further extend the range of doodles to any closed oriented surface. Uniqueness of minimal representatives is proved, and various examples of doodles are given with their minimal representatives. We also introduce the notion of virtual doodles, and show that there is a natural one-to-one correspondence between doodles on surfaces and virtual doodles on the plane.

On the entropy norm on the group of diffeomorphisms of closed oriented surface

We prove that the entropy norm on the group of diffeomorphisms of a closed orientable surface of positive genus is unbounded.

Covering diagrams over surface-knot diagrams

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained.

The flux homomorphism on closed hyperbolic surfaces and Anti-de Sitter three-dimensional geometry

Given a smooth spacelike surface ∑ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation p: π1 (S) → PSL2ℝ x PSL2ℝ where S is a closed oriented surface of genus ≥ 2, a canonical construction associates to ∑ a diffeomorphism φ∑ of S. It turns out that φ∑ is a symplectomorphism for the area forms of the two hyperbolic metrics h and h' on S induced by the action of p on ℍ2 x ℍ2. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that φ∑ is the composition of a Hamiltonian symplectomorphism of (S, h) and the unique minimal Lagrangian diffeomorphism from (S, h) to (S, h’).