145 research outputs found
Non-geometric veering triangulations
Recently, Ian Agol introduced a class of "veering" ideal triangulations for
mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the
singular points. These triangulations have very special combinatorial
properties, and Agol asked if these are "geometric", i.e. realised in the
complete hyperbolic metric with all tetrahedra positively oriented. This paper
describes a computer program Veering, building on the program Trains by Toby
Hall, for generating these triangulations starting from a description of the
homeomorphism as a product of Dehn twists. Using this we obtain the first
examples of non-geometric veering triangulations; the smallest example we have
found is a triangulation with 13 tetrahedra
Quantum Algorithms for Invariants of Triangulated Manifolds
One of the apparent advantages of quantum computers over their classical
counterparts is their ability to efficiently contract tensor networks. In this
article, we study some implications of this fact in the case of topological
tensor networks. The graph underlying these networks is given by the
triangulation of a manifold, and the structure of the tensors ensures that the
overall tensor is independent of the choice of internal triangulation. This
leads to quantum algorithms for additively approximating certain invariants of
triangulated manifolds. We discuss the details of this construction in two
specific cases. In the first case, we consider triangulated surfaces, where the
triangle tensor is defined by the multiplication operator of a finite group;
the resulting invariant has a simple closed-form expression involving the
dimensions of the irreducible representations of the group and the Euler
characteristic of the surface. In the second case, we consider triangulated
3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci
anyon model; the resulting invariant is the well-known Turaev-Viro invariant of
3-manifolds.Comment: 19 pages, 7 figure
Classification of genus-two surfaces in
We describe an algorithm to decide whether two genus-two surfaces embedded in
the 3-sphere are isotopic or not. The algorithm employs well-known techniques
in 3-manifolds topology, as well as a new algorithmic solution to a problem on
free groups.Comment: 65 pages, 22 figure
IST Austria Thesis
Algorithms in computational 3-manifold topology typically take a triangulation as an input and return topological information about the underlying 3-manifold. However, extracting the desired information from a triangulation (e.g., evaluating an invariant) is often computationally very expensive. In recent years this complexity barrier has been successfully tackled in some cases by importing ideas from the theory of parameterized algorithms into the realm of 3-manifolds. Various computationally hard problems were shown to be efficiently solvable for input triangulations that are sufficiently “tree-like.”
In this thesis we focus on the key combinatorial parameter in the above context: we consider the treewidth of a compact, orientable 3-manifold, i.e., the smallest treewidth of the dual graph of any triangulation thereof. By building on the work of Scharlemann–Thompson and Scharlemann–Schultens–Saito on generalized Heegaard splittings, and on the work of Jaco–Rubinstein on layered triangulations, we establish quantitative relations between the treewidth and classical topological invariants of a 3-manifold. In particular, among other results, we show that the treewidth of a closed, orientable, irreducible, non-Haken 3-manifold is always within a constant factor of its Heegaard genus
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