Related papers: Identifying lens spaces in polynomial time

We propose a solution to the problem of estimating a Riemannian metric associated with a given differentiable manifold. The metric learning problem is based on minimizing the relative volume of a given set of points. We derive the details…

We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to…

In this report we present an algorithm solving Triangle Counting in time $O(d^2n+m)$, where n and m, respectively, denote the number of vertices and edges of a graph G and d denotes its twin-width, a recently introduced graph parameter. We…

A Riemannian manifold is a called a good rational expander in dimension $i$ if every $i$-cycle bounds a rational $i+1$-chain of comparatively small volume. We construct 3-manifolds which are good expanders in all dimensions. On the other…

We discuss (arbitrary-dimensional) Lorentzian manifolds and the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. Recently, we have shown that in four dimensions a Lorentzian spacetime…

We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly…

We introduce an algorithm to decide isomorphism between tensors. The algorithm uses the Lie algebra of derivations of a tensor to compress the space in which the search takes place to a so-called densor space. To make the method practicable…

We determine lens surgeries (i.e.\ Dehn surgery yielding a lens space) along the $n$-twisted Whitehead link. To do so, we first give necessary conditions to yield a lens space from the Alexander polynomial of the link as: (1) $n=1$ (i.e.…

We investigate the problem of carving an $n$-face triangulated three-dimensional polytope using a tool to make cuts modelled by either a half-plane or sweeps from an infinite ray. In the case of half-planes cuts, we present a deterministic…

Given is a 1.5D terrain $\mathcal{T}$, i.e., an $x$-monotone polygonal chain in $\mathbb{R}^2$. For a given $2\le k\le n$, our objective is to approximate the largest area or perimeter convex polygon of exactly or at most $k$ vertices…

In recent years it has become popular to study machine learning problems in a setting of ordinal distance information rather than numerical distance measurements. By ordinal distance information we refer to binary answers to distance…

Let $M_n$ be a homology 3-sphere obtained by $\frac1n$-Dehn surgery along a $(p,q)$-torus knot. We consider a polynomial $\sigma_{(p,q,n)}(t)$ whose zeros are the inverses of the Reideimeister torsion of $M_n$ for…

Starting from the classical notion of an oriented congruence (i.e. a foliation by oriented curves) in $R^3$, we abstract the notion of an oriented congruence structure. This is a 3-dimensional CR manifold $(M,H, J)$ with a preferred…

Given an special type of triangulation $T$ for an oriented closed 3-manifold $M^3$ we produce a framed link in $S^3$ which induces the same $M^3$ by an algorithm of complexity $O(n^2)$ where $n$ is the number of tetrahedra in $T$ . The…

Given a set of $n$ points in the plane, and a parameter $k$, we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing $k$ points. We present the first near quadratic time algorithm for this…

Quantum invariants in low dimensional topology offer a wide variety of valuable invariants of knots and 3-manifolds, presented by explicit formulas that are readily computable. Their computational complexity has been actively studied and is…

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm…

For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of…

Evaluating a polynomial on a set of points is a fundamental task in computer algebra. In this work, we revisit a particular variant called trimmed multipoint evaluation: given an $n$-variate polynomial with bounded individual degree $d$ and…

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…