arXiv research

Defines Milnor number for foliations and shows its topological invariance.

3-manifolds study Hasse norm principle, akin to number fields.

We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…

The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.

Mazur, Kapranov, Reznikov, and others developed ``Arithmetic Topology,'' a theory describing some surprising analogies between 3-dimensional topology and number theory, which can be summarized by saying that knots are like prime numbers. We extend their work by proving several formulas concerning branched coverings of …

New method to decompose 4-manifolds with positive scalar curvature.

Study on Alexander polynomials in braids, linking number theory and topology.

We establish a number of results about smooth and topological concordance of knots in $S^1\times S^2$. The winding number of a knot in $S^1\times S^2$ is defined to be its class in $H_1(S^1\times S^2;\mathbb{Z})\cong \mathbb{Z}$. We show that there is a unique smooth concordance class of knots with winding number one. …

Study topological invariants of complexes for Riemannian manifolds.

Topology helps estimate chromatic numbers of random graphs on spheres.

Generically, topological insulators have conical points leading to Dirac-like currents.

The paper studies topological properties of Ricci shrinkers using weighted $L^2$ cohomology.

The paper characterizes and contrasts knots with high 4D clasp numbers.

We classify Legendrian knots of topological type $7_6$ having maximal Thurston--Bennequin number confirming the corresponding conjectures of Chongchitmate--Ng.

Paper introduces simplified formulas for Milnor's triple linking number.

Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In …

Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological r…

New spin on Hurwitz theory connects to Gromov-Witten theory and topological recursion.

Study Morse functions on projective plane using Reeb graphs.

In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules,…

We describe loss surfaces using topological Betti numbers.

Study classifies 3D Hessian manifolds, proving their topology.

The paper explores the topology of polygonal meshes and their properties.

Survey of Weber's class number problem and related topics.

The following numerical control over the topological equivalence is proved: two complex polynomials in $n\not= 3$ variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions $f_s \colon \mathbb{C}^n \to \mathbb{C}$ with isolated sin…

We show by example that the Chern numbers c_1^3 and c_1 c_2 of a complex 3-fold are not determined by the topology of the underlying smooth compact 6-manifold. In fact, we observe that infinitely many different values of a Chern number can be achieved by (integrable) complex structures on a fixed 6-manifold.

The paper introduces new invariants to study topological properties of map germs.

In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar …

Optimal curves minimize crossings on surfaces.

The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number, called the splitting graph. By using the splitting graph, we classify the embed…

Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.

It is well-known that classical two-dimensional topological field theories are in one-to-one correspondence with commutative Frobenius algebras. An important extension of classical two-dimensional topological field theories is provided by open-closed two-dimensional topological field theories. In this paper we extend o…

The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.

We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.

The paper sets limits on neural network sizes based on dataset shapes.

The Brasselet number of a function $f$ with nonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, we consider two function-germs $f,g:(X,0)\rightarrow(\mathbb{C},0)$ such that $f$ has isolated singularity at the origin and $g$ has a stratified one-dim…

New topological restrictions found for spaces with nonnegative Ricci curvature.

Focusing on a small set of proteins that i) fold in a concerted, all-or-none fashion and ii) do not contain knots or slipknots, we show that the Gauss linking integral, the torsion and the number of sequence-distant contacts provide information regarding the folding rate. Our results suggest that the global topology/ge…

The study of topological properties of random smooth maps, focusing on Kac-Rice formula and Betti numbers.

New harmonic functions show nodal sets can be topologically complex despite frequency and regularity constraints.

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular…

We introduce and study the Hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of fixed genus with specified numbers of backbones generated by s and chords generated by t. For the one-cut solution, the partition function, correlators and free energies are co…

Entropy of critical points generalizes Morse theory.

We use braids and linking number to explain why automobile shades fold into an odd number of loops.

We summarize the main results of our investigation of B-type topological Landau-Ginzburg models whose target is an arbitrary open Riemann surface. Such a Riemann surface need not be affine algebraic and in particular it may have infinite genus or an infinite number of Freudenthal ends. Under mild conditions on the Land…

We define a computable topological invariant $μ(γ)$ for generic closed planar regular curves $γ$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves witho…

We give topological lower bounds on the number of periodic and closed trajectories in strictly convex smooth billiards. We use variational reduction admitting a finite group of symmetries and apply topological approach based on equivariant Morse and Lusternik - Schnirelman theories. The paper continues results publishe…

Sampling random points can reveal submanifold topology.