Arnaud de Mesmay

On the complexity of immersed normal surfaces

Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our proof uses a reduction from Boolean constraint satisfaction problems where every variable appears in at most two clauses, using a classification theorem of Feder. We also investigate variants, and provide a polynomial-time algorithm to test for a local version of this problem.

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and vice-versa. This implies a conjecture by Przytycka and Przytycki (Graph structure theory. Contemporary Mathematics, vol. 147, 1993), a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov’s systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length

A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

O(g^{3/2}n^{1/2})

Testing graph isotopies on surfaces

O(g3/2n1/2) for any triangulated combinatorial surface of genus

Embeddability in

g

is NP-hard.

g with