Communications in Mathematics |

Editors: Jacob Mostovoy & Sergei Chmutov

It is conjectured since long that for any convex body $K \subset \mathbb{R}^n$ there exists a point in the interior of $K$ which belongs to at least $2n$ normals from different points on the boundary of $K$. The conjecture is known to be true for $n=2,3,4$. Motivated by a recent results of Y. Martinez-Maure, and an approach by A. Grebennikov and G. Panina, we prove the following: Let a compact smooth $m$-dimensional manifold $M^m$ be immersed in $ \mathbb{R}^n$. We assume that at least one of the homology groups $H_k(M^m,\mathbb{Z}_2)$ with $k<m$ vanishes. Then under mild conditions, almost every normal line to $M^m$ contains an intersection point of at least $\beta +4$ normals from different points of $M^m$, where $\beta$ is the sum of Betti numbers of $M^m$.

Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model represents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of $\mathbb Z^2$. Given a set $P$ of points in a compact convex domain $\Omega\subset \mathbb R^2$ this linearized model produces a tropical polynomial $G_P{\bf 0}_\Omega$. Here we present some quantitative statistical characteristics of this model and some speculative explanations. Namely, we study the dependence between the number $n$ of randomly dropped points $P=\{p_1,\dots,p_n\}\subset[0,1]^2=\Omega$ and the degree of the tropical polynomial $G_{P}{\bf 0}_\Omega$. We also study the distributions of the coefficients of $G_{P}{\bf 0}_\Omega$ and the correlation between them. This paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf 0}_\Omega)$ defined by $G_{P}{\bf 0}_\Omega$ is a small perturbation of the standard square grid lines. This explains a previously known fact that most of the edges of the tropical curve $C(G_{P}{\bf 0}_\Omega)$ are of directions $(1,0),(0,1),(1,1),(-1,1)$. The main theoretical result is that $C(G_{P}{\bf 0}_\Omega)\setminus (P\cap \partial\Omega)$, i.e. the tropical curve in $\Omega^\circ$ with marked points $P$ removed, is a tree.

A relaxation in the tropical sandpile model is a process of deforming a tropical hypersurface towards a finite collection of points. We show that, in the one-dimensional case, a relaxation terminates after a finite number of steps. We present experimental evidence suggesting that the number of such steps obeys a power law.

We study the problem of finding Young diagrams of maximum dimension, i. e. those with the largest number of Young tableaux of their shapes. Consider a class of Young diagrams that differ from a symmetric diagram by no more than one box $(i,j)$ in each row and column. It is proven that when moving boxes $(i,j), i>j$ to symmetric positions $(j,i)$, the original diagram is transformed into another diagram of the same size, but with a greater or equal dimension. A conjecture is formulated that generalizes the above fact to the case of arbitrary Young diagrams. Based on this conjecture, we developed an algorithm applied to obtain new Young diagrams of sizes up to 42 thousand boxes with large and maximum dimensions.

The notion of center of mass, which is very useful in kinematics, proves to be very handy in geometry (see [1]-[2]). Countless applications of center of mass to geometry go back to Archimedes. Unfortunately, the center of mass cannot be defined for sets whose total mass equals zero. In the paper we improve this disadvantage and assign to an n-dimensional affine space L over any field k the (n+1)-dimensional vector space over the field k of weighty points and mass dipoles in L. In this space, the sum of weighted points with nonzero total mass is equal to the center of mass of these points equipped with their total mass. We present several interpretations of the space of weighty points and mass dipoles in L, and a couple of its applications to geometry. The paper is self-contained and is accessible for undergraduate students.

We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For nonoriented simple graphs the definition is different, but for a certain class of graphs (namely, for intersection graphs of chord diagrams), it gives the same answer if we endow such a graph with an orientation induced by the chord diagram. We prove that this invariant satisfies Vassiliev's $4$-term relations and determines therefore a finite type knot invariant. We investigate the behaviour of the polynomial with respect to the Hopf algebra structure on the space of graphs and show that it takes a constant value on any primitive element in this Hopf algebra. We also provide a two-variable extension of the skew characteristic polynomial to embedded graphs and delta-matroids. The $4$-term relations for the extended polynomial prove that it determines a finite type invariant of multicomponent links.

We prove that the partial-dual genus polynomial considered as a function on chord diagrams satisfies the four-term relation. Thus it is a weight system from the theory of Vassiliev knot invariants.

Lando framed graph bialgebra is generated by framed graphs modulo 4-term relations. We provide an explicit set of generators of its primitive subspace and a description of the set of relations between the generators. We also define an operation of leaf addition that endows the primitive subspace of Lando algebra with a structure of a module over the ring of polynomials in one variable and construct a 4-invariant that satisfies a simple identity with respect to the vertex-multiplication.

We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.

Recently, Chmutov proved that the partial-dual polynomial considered as a function on chord diagrams satisfies the four-term relations. In this paper, we show that this function on framed chord diagrams also satisfies the four-term relations, i.e., is a framed weight system.