Tight triangulations of closed 3-manifolds

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the K\"{u}hnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $\mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $\mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $\mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $\mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $\beta_1$, then $(n-4)(617n- 3861) \leq 15444\beta_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.Comment: 21 pages, 1 figur

Computing Bounded Chains and Surfaces in a Simplicial Complex with Bounded-treewidth 1-skeleton

We consider three problems on simplicial complexes: the Optimal Bounded Chain Problem, the Optimal Homologous Chain Problem, and 2-Dim-Bounded-Surface. The Optimal Bounded Chain Problem asks to find the minimum weight d-chain in a simplicial complex K bounded by a given (d−1)-chain, if such a d-chain exists. The Optimal Homologous Chain problem asks to find the minimum weight (d−1)-chain in K homologous to a given (d−1)-chain. 2-Dim-Bounded-Surface asks whether or not there is a subcomplex of K homeomorphic to a given compact, connected surface bounded by a given subcomplex. All three problems are NP-hard, and the first two problems are hard to approximate within any constant factor assuming the Unique Games Conjecture. We prove that all three problems are fixed-parameter tractable with respect to the treewidth of the 1-skeleton of K

Efficient algorithms to decide tightness

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is “as convex as possible”, given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but more efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces. In this article, we present a new polynomial time procedure to decide tightness for triangulations of 3-manifolds – a problem which previously was thought to be hard. In addition, for the more difficult problem of deciding tightness of 4-dimensional combinatorial manifolds, we describe an algorithm that is fixed parameter tractable in the treewidth of the 1-skeletons of the vertex links. Finally, we show that simpler treewidth parameters are not viable: for all non-trivial inputs, we show that the treewidths of both the 1-skeleton and the dual graph must grow too quickly for a standard treewidth-based algorithm to remain tractable