Related papers: A cube Slice that is not a Zonoid
Let p be prime. A noncommutative p-solenoid is the C*-algebra of Z[1/p] x Z[1/p] twisted by a multiplier of that group, where Z[1/p] is the additive subgroup of the field Q of rational numbers whose denominators are powers of p. In this…
A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all…
We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.
We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of…
The cut polytope ${\rm CUT}(n)$ is the convex hull of the cut vectors in a complete graph with vertex set $\{1,\ldots,n\}$. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation…
A subgraph of the $n$-dimensional hypercube is called 'layered' if it is a subgraph of a layer of some hypercube. In this paper we show that there exist subgraphs of the cube of arbitrarily large girth that are not layered. This answers a…
Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation…
A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation…
We study the volume of central hyperplane sections of the cube. Using Fourier analytic and variational methods, we retrieve a geometric condition characterizing critical sections which, by entirely different methods, was recently proven by…
Solenoids induced by split sequences are introduced, as the inverse limit object of a sequence of fold maps. The topology of a solenoid is explored, and it is established that solenoids have naturally arising singular foliated structures.…
The definition of the centroid in finite dimensions does not apply in a function space because of the lack of a translation invariant measure. Another approach, suggested by Nik Weaver, is to use a suitable collection of finite-dimensional…
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…
We study the geometry, Hodge theory and derived category of cubic fourfolds containing several planes and their associated twisted K3 surfaces. We focus on the case of two planes intersecting along a line.
Consider a $d$-dimensional closed ball $B$ whose center coincides with that of the hypercube $[0,1]^d$. Pick the radius of $B$ in such a way that the vertices of the hypercube are outside of $B$ and the midpoints of its edges in the…
The paper deals with cubic 1-variable polynomials whose Julia sets are connected. Fixing a bounded type rotation number, we obtain a slice of such polynomials with the origin being a fixed Siegel point of the specified rotation number. Such…
Ribbon decomposition is a way to obtain a surface with boundary (compact, not necessarily oriented) from a collection of disks by joining them with narrow ribbons attached to segments of the boundary. Counting ribbon decompositions gives…
A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and…
We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed. We also prove that, for infinitely many integers $k$, the largest possible diameter of a lattice zonotope contained in the…
A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are…
The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the classical signature function of the knot to give a new lower…