Wagner, UliIST Austria ; Spreer, Jonathan
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.
Huszár K. Combinatorial Width Parameters for 3-Dimensional Manifolds. IST Austria; 2020. doi:10.15479/AT:ISTA:8032
Huszár, K. (2020). Combinatorial width parameters for 3-dimensional manifolds. IST Austria. https://doi.org/10.15479/AT:ISTA:8032
Huszár, Kristóf. Combinatorial Width Parameters for 3-Dimensional Manifolds. IST Austria, 2020. https://doi.org/10.15479/AT:ISTA:8032.
K. Huszár, Combinatorial width parameters for 3-dimensional manifolds. IST Austria, 2020.
Huszár K. 2020. Combinatorial width parameters for 3-dimensional manifolds, IST Austria, xviii+120p.
Huszár, Kristóf. Combinatorial Width Parameters for 3-Dimensional Manifolds. IST Austria, 2020, doi:10.15479/AT:ISTA:8032.
Kristof_Huszar-Thesis.pdf 2.64 MB
Kristof_Huszar-Thesis-source.zip 7.16 MB