arXiv subjects

Hyam Rubinstein

Publications and source records attributed to Hyam Rubinstein.

5 recordsLinked to original sources

arXiv2018

On minimal ideal triangulations of cusped hyperbolic 3-manifolds

Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional $\mathbb{Z}_2$-cohomology. The underlying blueprint is now used in the study of minimal ideal triangulations. As an application, it is shown that the monodromy ideal triangulations of the hyperbolic once-punctured torus bundles are minimal.

math.GT↗

arXiv2017

Non-uniqueness of the homotopy class of bounded curvature paths

A bounded curvature path is a continuously differentiable piecewise $C^2$ path with a bounded absolute curvature that connects two points in the tangent bundle of a surface. In this work, we analyze the homotopy classes of bounded curvature paths for points in the tangent bundle of the Euclidean plane. We show the existence of connected components of bounded curvature paths that do not correspond to those under (regular) homotopies obtaining the first results in the theory outside optimality. An application to robotics is presented.

math.MG↗

arXiv2016

The 3D-index and normal surfaces

Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating series of normal surfaces associated to the ideal triangulation. This shows a connection of the 3D-index with classical normal surface theory, and fulfills a dream of constructing topological invariants of 3-manifolds using normal surfaces.

math.GT↗

arXiv2013

Four-dimensional Haken cobordism theory

Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $π_1(W)$ is injective. In this paper we prove that there are 4-dimensional Haken cobordisms whose boundary consists of any two closed Haken 3-manifolds. In particular, each closed Haken 3-manifold is the $π_1$-injective boundary of some Haken 4-manifold.

math.GT↗

arXiv2011

Mapping class groups of Heegaard splittings

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the surface onto itself, modulo isotopies that keep the surface on itself. We characterize the mapping classes that restrict to periodic and reducible automorphisms of the surface.

math.GT↗