arXiv research

Algorithm identifies spheres with maximal Buchstaber number.

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The…

Study of universal complexes in toric topology with applications in category theory.

LVM and LVMB manifolds are a large family of examples of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a very natural action of the real torus and the quotient of this action is a simple polytope. This quotient allows us to relate c…

LVM and LVMB manifolds are a large family of examples of non kahler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a very natural action of the real torus and the quotient of this action is a simple polytope. This quotient allows us to relate c…

We discuss the relation between arc index, maximal Thurston--Bennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal Thurston--Bennequin number for all knots with at most 11 crossings. For some of these knots, the calculation requires a consideration of cables which a…

It is natural to expect and simple to prove that every conformally flat space possess the maximal number of conformal Killing vector fields (CKVs). On the other hand, it is interesting to ask whether the converse is true. Is conformal flatness a necessary condition for the existence of the maximal number of CKVs? In th…

For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand toru…

We prove two conjectures of C. Gordon. We show that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10, and that the maximal intersection number between exceptional slopes is 8. The proof uses a combination of new geometric techniques and a rigorous computer-assisted calculation.

We compute the maximal Thurston-Bennequin number for a Legendrian two-bridge knot or oriented two-bridge link in standard contact R^3, by showing that the upper bound given by the Kauffman polynomial is sharp. As an application, we present a table of maximal Thurston-Bennequin numbers for prime knots with nine or fewer…

We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2^$ for graphs with n+1 singularities. We also give an explicit description of the family of entire maximal graphs with a f…

Proves existence of maps with arbitrary ends and conditions for maxfaces.

We give lower bounds on the maximal injectivity radius for a closed orientable hyperbolic 3-manifold M with first Betti number 2, under some additional topological hypotheses. A corollary of the main result is that if M has first Betti number 2 and contains no fibroid surface then its maximal injectivity radius exceeds…

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

Flat open manifolds with full first Betti number have zero curvature.

It is known that the maximal homological degree of the Khovanov homology of a knot gives a lower bound of the minimal positive crossing number of the knot. In this paper, we show that the maximal homological degree of the Khovanov homology of a cabling of a knot gives a lower bound of the minimal positive crossing numb…

Noncompact RCD spaces with maximal first Betti number are rigid.

We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.

Fast algorithms developed for adaptive and fully adaptive submodular maximization problems.

In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. In t…

Unique maximal curve systems found for up to 5 punctures.

For a fixed compact Riemann surface X, of genus at least 2, we count the number of connected components of the moduli space of maximal Higgs bundles over X for the hermitian groups $PSp(2n,R)$, $PSO^*(2n)$, $PSO_0(2,n)$ and $E_6^{-14}$. Hence the same result follows for the number of connected components of the moduli …

New invariant distinguishes Legendrian surfaces in 5-manifolds.

In Bipartite Correlation Clustering (BCC) we are given a complete bipartite graph $G$ with `+' and `-' edges, and we seek a vertex clustering that maximizes the number of agreements: the number of all `+' edges within clusters plus all `-' edges cut across clusters. BCC is known to be NP-hard. We present a novel approx…

We give examples showing that Kidwell's inequality for the maximal degree of the Brandt-Lickorish-Millett-Ho polynomial is in general not sharp.

Study classifies twist knots with maximal self-linking number in S^3.

Maximal rank Coxeter quotients found for 1.7M knots up to 16 crossings.

A maximally linkless graph is a graph that can be embedded in $\mathbb{R}^3$ without any links, but cannot be embedded in such a way if any other edge is added to the graph. Recently, a family of maximally linkless graphs was found with $m=3n-3$ edges. We improve upon this by demonstrating a new family of maximally lin…

Extends flat submanifold properties from hyperbolic plane to symmetric spaces.

We prove a lower bound on the number of maximally broken trajectories of the negative gradient flow of a Morse-Smale function on a closed aspherical manifold in terms of integral (torsion) homology.

The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling, therefore its Thurston-Bennequin number is maximal among Legendrian …

Poly-view contrastive learning improves image representation learning.

We show that the number of noncommensurable lattices, hence also that of maximal lattices in SO(1,n) is at least exponential. To do so we construct large families of noncommensurable hybrid hyperbolic (Gromov/Piatetski-Shapiro) manifolds.

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More gene…

The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We…

Scientific explanation often requires inferring maximally predictive features from a given data set. Unfortunately, the collection of minimal maximally predictive features for most stochastic processes is uncountably infinite. In such cases, one compromises and instead seeks nearly maximally predictive features. Here, …

The language of maximal lexicographic representatives of elements in the positive braid monoid $A_n$ with $n$ generators is a regular language. We describe with great detail the smallest Finite State Automaton accepting such language, and study the proportion of elements of length $k$ whose maximal lexicographic repres…

Let M be a closed, irreducible, genus two 3-manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M. Then each component manifold M_i of M-F has handle number at most one, i.e. admits a Heegaard splitting obtained by attaching a single 1-handle to one or two …

A new algorithm learns MAGs from data more efficiently using entropy.

MONSTOR estimates influence in unseen networks with high accuracy.

The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and wea…

In this paper we investigate a new class of growth rate maximization problems based on impulse control strategies such that the average number of trades per time unit does not exceed a fixed level. Moreover, we include proportional transaction costs to make the portfolio problem more realistic. We provide a Verificatio…

We analyze the frequency spectrum of quantum neural networks using algebraic methods and prove maximality results.

Study on invariant anti-quasi-Sasakian structures on compact manifolds.

Greedy policy maximizes information in unknown linear systems.

A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in $ł^3.$ In addition, $\sb$ is called of finite type if it has finite topology, finitely many singular points and $\tilde{\sb}$ is finitely …

New algorithm for maximizing submodular functions in real-time data changes.

The paper calculates the Δ-unknotting number for positive pretzel knots.