Related papers: k-Normal surfaces
We define a 2-normal surface to be one which intersects every 3-simplex of a triangulated 3-manifold in normal triangles and quadrilaterals, with one or two exceptions. The possible exceptions are a pair of octagons, a pair of unknotted…
In this thesis, we use normal surface theory to understand certain properties of minimal triangulations of compact orientable 3-manifolds. We describe the collapsing process of normal 2-spheres and disks. Using some geometrical…
We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with non-empty boundary which has a quadrilateral in each tetrahedron. While this condition…
In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have…
This} paper presents relations between least area and normal surfaces, embedded in either a Euclidean or hyperbolic $3$-manifold. A relaxed version of normal surfaces, termed quasi-normal, is introduced, and it is shown that under…
Let M be the interior of a compact 3-manifold with non-empty boundary, and T be an ideal (topological) triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi-angle structures…
We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of…
The concept of a normal surface in a triangulated, compact 3-manifold was generalised by Thurston to a spun-normal surface in a non-compact 3-manifold with ideal triangulation. This paper defines a boundary curve map which takes a…
This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact, orientable 3--manifold in terms of the quadrilaterals in its cell decomposition---different bounds arise from varying hypotheses on the surface…
We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a…
It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct…
A well-known result of Walsh states that if $\mathcal T^*$ is an ideal triangulation of an atoroidal, acylindrical, irreducible, compact 3-manifold with torus boundary components, then every properly embedded, two-sided, incompressible…
The formula for the Turaev-Viro invariant of a 3-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. This paper analyzes convergence of this sum when t does not lie on the unit…
A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to…
We show that a strongly irreducible and boundary-strongly irreducible surface can be isotoped to be almost normal in a triangulated 3-manifold.
The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.
We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…
We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting…
Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by…
A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no…