arXiv research

Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.

This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth famil…

Semi-Equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the $2$-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-eq…

Algorithm determines spatial graph isomorphism with vertex, edge colorings and orientations.

Automorphisms of fine 1-curve graph linked to surface homeomorphisms.

Graph alignment in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact information-theoretic threshold for graph alignment in correlated Erdős-Rényi graphs. However, very little is known about the existence of e…

A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible triangulations of any punctured surface is established. Complete lists of irreducible tria…

We study triangulations $\cal T$ defined on a closed disc $X$ satisfying the following condition: In the interior of $X$, the valence of all vertices of $\cal T$ except one of them (the irregular vertex) is $6$. By using a flat singular Riemannian metric adapted to $\cal T$, we prove a uniqueness theorem when the valen…

Classifies 2-uniform maps on torus with formulas and asymptotic bounds.

Using the link surgery formula for Heegaard Floer homology we find a spectral sequence from the lattice homology of a plumbing tree to the Heegaard Floer homology of the corresponding 3-manifold. This spectral sequence shows that for graphs with at most two "bad" vertices, the lattice homology is isomorphic to the Heeg…

We propose a representation of graph as a functional object derived from the power iteration of the underlying adjacency matrix. The proposed functional representation is a graph invariant, i.e., the functional remains unchanged under any reordering of the vertices. This property eliminates the difficulty of handling e…

Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.

The article explores symmetric maps on surfaces, focusing on semi-equivelar maps.

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give…

To each oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The obtained transformation L is the main object of study of the present paper. We pose a problem on the inversion of the transformation L. We shall show that this problem is closely rela…

Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.

Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.

Paper solves graph matching for correlated Erdős--Rényi graphs.

For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between "princ…

A triangulated $d$-manifold $K$, satisfies the inequality $\binom{f_0(K)-d-1}{2}\geq \binom{d+2}{2}β_1(K;\mathbb{Z}_2)$ for $d\geq 3$. The triangulated $d$-manifolds that meet the bound with equality are called {\em tight neighborly}. In this paper, we present tight neighborly triangulations of 4-manifolds on 15 vertic…

New neural networks learn graph symmetries.

Minimal triangulations of spheres map almost linearly to boundaries of high-dimensional polytopes.

Link Floer homology is split into snake complexes and local systems.

Theory of T-duality for transitive Courant algebroids developed.

Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.

Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all…

This note is about the geometry of the pants graph P(S), a natural simplicial graph associated to a finite type topological surface S where vertices represents pants decompositions. The main result in this note ascserts that for a multicurve Q whose complement is a number of subsurfaces of complexity at most 1. We prov…

For the cotangent bundle $T^{*}K$ of a compact Lie group $K$, we study the complex-time evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space $L^{2}(K)$ under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state tra…

Assume that Γ_{v_0} is a tree with vertex set Vert(Γ_{v_0})={v_0, v_1,..., v_n}, and with an integral framing (weight) attached to each vertex except v_0. Assume furthermore that the intersection matrix of G=Γ_{v_0}-{v_0} is negative definite. We define a filtration on the chain complex computing the lattice homology o…

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale f…

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel an…

For a half-translation surface (S,q), the associated saddle connection complex A(S,q) is the simplicial complex where vertices are the saddle connections on (S,q), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. …

In this paper we prove the knight move theorem for the chromatic graph cohomologies with rational coefficients introduced by L. Helme-Guizon and Y. Rong. Namely, for a connected graph G with n vertices the only non-trivial cohomology groups $H^{i,n-i}(G)$, $H^{i,n-i-1}(G)$ come in isomorphic pairs: $H^{i,n-i}(G)\cong H…

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that…

Study semi-coarse spaces' homotopy and homology, extending coarse geometry.

New hypergraph neural network learns variable-sized hyperedges.

For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $ι: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms of dg-manifolds. The vertical tangent bundle $T^p\mathcal M$ then inherit…

The paper finds and visualizes unique geometric polyhedra and tori with few vertices.

For integers $d \geq 2$ and $ε= 0$ or 1, let $S^{1, d - 1}(ε)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $ε= 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $ε= 1$. The main results of this paper are: (a) if $d \equiv ε$ (mod 2) then $S^{1, d - 1}(ε)$ has a unique minimal triangulation using $2d + 3$ …

A geodesic net with 4 boundary vertices and 25 balanced vertices is constructed.

This paper classifies semi-equivelar maps on a special surface.

We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation…

Graph kernels based on the $1$-dimensional Weisfeiler-Leman algorithm and corresponding neural architectures recently emerged as powerful tools for (supervised) learning with graphs. However, due to the purely local nature of the algorithms, they might miss essential patterns in the given data and can only handle binar…

In this paper we construct and study a new 15-vertex triangulation $X$ of the complex projective plane $\CP^2$. The automorphism group of $X$ is isomorphic to $S_4\times S_3$. We prove that the triangulation $X$ is the minimal by the number of vertices triangulation of $\CP^2$ admitting a chess colouring of four-dimens…

Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.

We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product $S^2\times S^1$ and 615 triangulations of the twisted sphere product $S^2_\times_S^1$. All the 3-spheres…

The string vertices of closed string field theory are subsets of the moduli spaces of punctured Riemann surfaces that satisfy a geometric version of the Batalin-Vilkovisky master equation. We present a homological proof of existence of string vertices and their uniqueness up to canonical transformations. Using hyperbol…

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one full turn of the …