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 atleast $2n$ normals from different points on the boundary of $K$. The conjectureis 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 thatat 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 differentpoints of $M^m$, where $\beta$ is the sum of Betti numbers of $M^m$.

Tropical sandpile model (or linearized sandpile model) is the only knowncontinuous geometric model exhibiting self-organised criticality. This modelrepresents the scaling limit behavior of a small perturbation of the maximalstable sandpile state on a big subset of $\mathbb Z^2$. Given a set $P$ ofpoints in a compact convex domain $\Omega\subset \mathbb R^2$ this linearizedmodel produces a tropical polynomial $G_P{\bf 0}_\Omega$. Here we present some quantitative statistical characteristics of this modeland some speculative explanations. Namely, we study the dependence between thenumber $n$ of randomly dropped points$P=\{p_1,\dots,p_n\}\subset[0,1]^2=\Omega$ and the degree of the tropicalpolynomial $G_{P}{\bf 0}_\Omega$. We also study the distributions of thecoefficients of $G_{P}{\bf 0}_\Omega$ and the correlation between them. Thispaper's main (experimental) result is that the tropical curve $C(G_{P}{\bf0}_\Omega)$ defined by $G_{P}{\bf 0}_\Omega$ is a small perturbation of thestandard square grid lines. This explains a previously known fact that most ofthe 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 atropical hypersurface towards a finite collection of points. We show that, inthe one-dimensional case, a relaxation terminates after a finite number ofsteps. We present experimental evidence suggesting that the number of suchsteps 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 aclass of Young diagrams that differ from a symmetric diagram by no more thanone 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 istransformed into another diagram of the same size, but with a greater or equaldimension. A conjecture is formulated that generalizes the above fact to thecase of arbitrary Young diagrams. Based on this conjecture, we developed analgorithm applied to obtain new Young diagrams of sizes up to 42 thousand boxeswith large and maximum dimensions.

The notion of center of mass, which is very useful in kinematics, proves tobe very handy in geometry (see [1]-[2]). Countless applications of center ofmass to geometry go back to Archimedes. Unfortunately, the center of masscannot be defined for sets whose total mass equals zero. In the paper weimprove this disadvantage and assign to an n-dimensional affine space L overany field k the (n+1)-dimensional vector space over the field k of weightypoints and mass dipoles in L. In this space, the sum of weighted points withnonzero total mass is equal to the center of mass of these points equipped withtheir total mass. We present several interpretations of the space of weightypoints and mass dipoles in L, and a couple of its applications to geometry. Thepaper is self-contained and is accessible for undergraduate students.

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

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

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