arXiv research

Lean 4 formalizes Stokes' theorem for smooth singular cubes.

We smooth the singularities of a strictly hyperbolized smooth cube manifold.

The paper defines grid homologies for singular links in lens spaces and constructs a resolution cube for knot Floer homology.

We prove that smooth cube manifolds have normal smooth structures.

We prove that strictly hyperbolized smooth cube manifolds admit normal smooth structures.

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the ho…

Study symmetries in smoothed polygonal links.

Smooth isotopy on cube saves energy with extra dimensions.

We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.

Paper extends cobordism maps in Khovanov and instanton homologies.

Researchers solve 3D cube complex boundary rigidity problem.

In a 2013 paper, Gromov proves that if smooth Riemannian metrics $g_i$ converge to a smooth Riemannian metric $g$ uniformly, and $g_i$ have scalar curvature uniformly bounded below, then $g$ shares the same scalar curvature lower bound. In some places in the paper, the proofs are only sketched. In this paper we explain…

New method approximates hyperbolic lattices using cube complexes.

Defines cobordism maps connecting Khovanov and instanton homologies.

This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen…

Paper proves Gromov's cube inequality in all dimensions.

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells…

We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three positive numbers $a_1, a_2, a_3$. Our interest is to determine the type of sing…

Simplified and extended a method for rearranging infinite configurations of cubes.

Generalizes Leighton's theorem to cube complexes.

In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Fl…

The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex $C_{F}(S)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is isomorphic to the HOMFLY-PT homology of $S$. Together with a naturality condition on t…

Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corres…

New geometric spine for Artin groups defined by cube complexes.

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. We will show that the cube number detects chirality in all cases computed thus far, and distinguishes certain legendrian knots.

Study mapping class groups on CAT(0) cube complexes.

We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a "signed BW cubed-complex", and give a characterization for a signed BW cubed-complex to have the underlying space which is homeomorphic to an alternating link exterior.

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non…

Rep-tiles fill cubes in any dimension.

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…

Finite stature proven for cube complexes with cyclonormal edges.

Cube edges curves minimize systole length.

This paper examines the category C^k_{d,n} whose morphisms are d-dimensional smooth manifolds that are properly embedded in the product of a k-dimensional cube with an (d+n-k)-dimensional Euclidean space. There are k directions to compose k-dimensional cubes, so C^k_{d,n} is a (strict) k-tuple category. The geometric r…

We demonstrate the homogeneity of the Hilbert Cube. In particular, we construct explicit self-homeomorphisms of the Hilbert cube so that given any two points, a homeomorphism moving one to the other may be realized.

Beside simplices, $n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter $n$-cubes are not classified. We show that there is no hyperbolic Coxeter $n$-cube for $n\geq~6$, and provide a full classification for $n\leq 5$. Our methods, which are essentially of combin…

In this short note we highlight some of the differences between cube diagrams and grid diagrams. We also list examples of small cube diagrams for all knots up to 7 crossings and give some examples of links.

Groups on CAT(0) cube complexes grow exponentially uniformly.

The $n$-skeleton of the canonical cubulation $\cal C$ of $\mathbb{R}^{n+2}$ into unit cubes is called the {\it canonical scaffolding} ${\cal{S}}$. In this paper, we prove that any smooth, compact, closed, $n$-dimensional submanifold of $\mathbb{R}^{n+2}$ with trivial normal bundle can be continuously isotoped by an amb…

CUBE explains models by balanced experiments and contrasts.

We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.

Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.

New group acts on complex but not in lower dimensions.

We count orientable small covers over cubes. We also get estimates for $O_n/R_n$, where $O_n$ is the number of orientable small covers and $R_n$ is the number of all small covers over an $n$-cube up to the Davis-Januszkiewicz equivalence.

Extends folding techniques to study subgroups of CAT(0) cube complexes.

Study smoothings of singular intersections of ellipsoids.

The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.

We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation of real trees. Our main result states that a finitely generated group $G$ acts n…

We combine ideas of Scott and Swarup on good position for almost invariant subsets of a group with ideas of Sageev on constructing cubings from such sets. We construct cubings which are more canonical than in Sageev's original construction. We also show that almost invariant sets can be chosen to be in very good positi…