Normal Surface Theory Research Articles

On the hardness of finding normal surfaces

For many fundamental problems in computational topology, such as unknot recognition and 3-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms for these problems is normal surface theory. However, we currently have a poor understanding of the computational complexity of problems in normal surface theory: many such problems are still not known to have polynomial-time algorithms, yet proofs of \(\mathrm {NP}\)-hardness also remain scarce. We give three results that provide some insight on this front. A number of modern normal surface theoretic algorithms depend critically on the operation of finding a non-trivial normal sphere or disc in a 3-dimensional triangulation. We formulate an abstract problem that captures the algebraic and combinatorial aspects of this operation, and show that this abstract problem is \(\mathrm {NP}\)-complete. Assuming \(\mathrm {P}\ne \mathrm {NP}\), this result suggests that any polynomial-time procedure for finding a non-trivial normal sphere or disc will need to exploit some geometric or topological intuition. Another key operation, which applies to a much wider range of topological problems, involves finding a vertex normal surface of a certain type. We study two closely-related problems that can be solved using this operation. For one of these problems, we give a simple alternative solution that runs in polynomial time; for the other, we prove \(\mathrm {NP}\)-completeness.

Efficient triangulations and boundary slopes

For a compact, orientable, irreducible, ∂-irreducible, and an-annular 3-manifold, it is shown there are only finitely many boundary slopes for incompressible and ∂-incompressible surfaces of a bounded Euler characteristic. We use normal surface theory and the inverse relationship of crushing a triangulation along a normal surface [8] and that of inflating an ideal triangulation [12] to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. It is shown for a compact 3-manifold with boundary, satisfying these topological conditions, any triangulation can be modified to a boundary-efficient triangulation; furthermore, it can be decided if a triangulation of such a manifold is boundary-efficient.

Determining hyperbolicity of compact orientable 3-manifolds with torus boundary

Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.

Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof, with several variations at each stage, that uses only the statement of the geometrization theorem, basic hyperbolic geometry, and old results from combinatorial topology and computer science. For this result, we do not rely on normal surface theory, methods from geometric group theory, nor methods used to prove geometrization. The new result is that the homeomorphism problem is elementary recursive, i.e., that the computational complexity is bounded by a bounded tower of exponentials. This result relies on normal surface theory, Mostow rigidity, and bounds on the computational complexity of solving algebraic equations.

From Normal Surfaces to Normal Curves to Geodesics on Surfaces

This paper gives a study of a two dimensional version of the theory of normal surfaces; namely, a study o normal curves and their relations with respect to geodesic curves. This study results with a nice discrete approximation of geodesics embedded in a triangulated orientable Riemannian surface. Experimental results of the two dimensional case are given as well.

Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

Abstract The colorful simplicial depth of a collection of $d+1$ finite sets of points in Euclidean $d$-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. [7]. Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton [6] in the theory of normal surfaces.

Bounds for the genus of a normal surface

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 or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.

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.

The 3D-index and normal surfaces

Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating series of normal surfaces associated to the ideal triangulation. This shows a connection of the 3D-index with classical normal surface theory, and fulfills a dream of constructing topological invariants of 3-manifolds using normal surfaces.

Core curves of triangulated solid tori

One of the most powerful tools in 3-manifold topology is normal surface theory. Many topologically relevant surfaces can be placed into normal form (or some variant of this) with respect to any triangulation T of the manifold M . An important case of this phenomenon is when M is the solid torus and the surface is a meridian disc. The existence of a meridian disc in normal form and the fact that this can be algorithmically detected is the key to Haken’s solution to the problem of recognizing the unknot [1]. However, normal surface theory suffers from some substantial limitations, possibly the most important of which is that every normal meridian disc may have ‘exponential complexity’. More precisely, if n is the number of tetrahedra in T , then normal surface theory only produces a meridian disc D with at most 2 normal triangles and squares, where k = 10 (see [2]). This is more than just an artifice of the theory, because one can find triangulations of the solid torus where every normal meridian disc has exponentially many squares and triangles. Indeed, we will do this explicitly in Section 6. However, in this paper, we show that for a related problem, there is a solution with linearly bounded complexity. In addition to the meridian disc, the solid torus contains another important sub-object: the core curve C, which is defined, up to ambient isotopy, to be {∗} × S ⊂ D × S, where ∗ is a point in the interior of D. In this paper, we will address the problem of placing C into ‘normal form’. The surprising conclusion is that this can be achieved with a linear upper bound on the ‘complexity’ of C.

Incompressibility criteria for spun-normal surfaces

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 nonempty boundary which has a quadrilateral in each tetrahedron. While this condition is far from being necessary, it is powerful enough to give two new results: the existence of alternating knots with noninteger boundary slopes, and a proof of the Slope Conjecture for a large class of 2-fusion knots. While the condition and conclusion are purely topological, the proof uses the Culler-Shalen theory of essential surfaces arising from ideal points of the character variety, as reinterpreted by Thurston and Yoshida. The criterion itself comes from the work of Kabaya, which we place into the language of normal surface theory. This allows the criterion to be easily applied, and gives the framework for proving that the surface is incompressible. We also explore which spun-normal surfaces arise from ideal points of the deformation variety. In particular, we give an example where no vertex or fundamental surface arises in this way.

Essential disks and semi-essential surfaces in 3-manifolds

If M is a manifold with compressible boundary, we analyze essential disks in M, as well as incompressible, but not necessarily ∂-incompressible, surfaces in M. We are most interested in the case where M is a handlebody or compression body. The analysis depends on a new normal surface theory and is analogous to the analysis of incompressible surfaces using branched surfaces. The surfaces in question are carried by a finite set of “carriers.” We hope our theory of carriers will be used in other papers to describe objects representing limits of essential disks in a handlebody or in a 3-manifold having compressible boundary.

Converting between quadrilateral and standard solution sets in normal surface theory

The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.

Angle structures and normal surfaces

Let M be the interior of a compact 3-manifold with boundary, and let T be an ideal triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi-angle structures and generalised angle structures on (M; T ) respectively in terms of a generalised Euler characteristic function on the solution space of the normal surface theory of (M; T ). This extends previous work of Kang and Rubinstein, and is itself generalised to a more general setting for 3-dimensional pseudo-manifolds.

Algorithms for finding proper essential surfaces in 3-manifolds

In this paper, we present an algorithm which, for a given compact orientable irreducible boundary irreducible 3-manifold M, verifies whether M contains an essential orientable surface (possibly, with boundary), whose genus is at most N. The algorithm is based on Haken’s theory of normal surfaces, and on a trick suggested by Jaco and consisting in estimating the mean length of boundary curves in an unknown essential surface of a given genus in the given manifold.

NORMAL SURFACES IN THE FIGURE-8 KNOT COMPLEMENT

An approach to normal surface theory for non-compact 3-manifolds with respect to ideal pseudo-triangulations is described in [12]. The figure-8 knot complement with a given ideal pseudo-triangulation with two ideal tetrahedra is fully worked out. We construct all Q-fundamental surfaces and the remaining normal surfaces are obtained by linear geometric sums of these. We also construct all almost normal surfaces in the figure-8 knot complement. As a result we obtain an example which does not contain any normal or almost normal surface representing a minimal Seifert surface of the knot. This leads that the figure-8 knot is fibered.

The canonical decomposition of once-punctured torus bundles

We determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defined by Floyd and Hatcher. Unlike previous work on this problem, the techniques we use are not geometric. Instead, they involve angled polyhedral decompositions, thin position and a version of almost normal surface theory.

Decision problems in the space of Dehn fillings

Normal surface theory is used to study Dehn fillings of a knot-manifold. We use that any triangulation of a knot-manifold may be modified to a triangulation having just one vertex in the boundary. In this situation, it is shown that there is a finite computable set of slopes on the boundary of the knot-manifold, which come from boundary slopes of normal or almost normal surfaces. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely those manifolds obtained by Dehn filling of a given knot-manifold that: (1) are reducible, (2) contain two-sided incompressible surfaces, (3) are Haken, (4) fiber over S 1, (5) are the 3-sphere, and (6) are a lens space. Each of these algorithms is a finite computation. Moreover, in the case of essential surfaces, we show that the topology of each filled manifold is strongly reflected in the combinatorial properties of a triangulation of the knot-manifold with just one vertex in the boundary. If a filled manifold contains an essential surface then the knot-manifold contains an essential normal vertex solution which caps off to an essential surface of the same type in the filled manifold. (Normal vertex solutions are the premier class of normal surface and are computable.)

An Algorithm for Acylindrical Surfaces in 3-manifolds

An algorithm to decide if an orientable atoroidal 3-manifold contains closed incompressible acylindrical surfaces, and construct closed incompressible acylindrical surfaces is given. Mainly, the normal surface theory is used. To assure that the algorithm stops after finite steps, we show that each acylindrical surface is isotopic to some ``edge surface'' which is constructible.

Heegaard splittings with boundary and almost normal surfaces

This paper generalizes the definition of a Heegaard splitting to unify the concepts of thin position for 3-manifolds [M. Scharlemann, A. Thompson, Contemp. Math., Vol. 164, Amer. Math. Soc., 1994, pp. 231–238], thin position for knots [D. Gabai, J. Differential Geom. 26 (1987) 479–536], and normal and almost normal surface theory [W. Haken, Acta Math. 105 (1961) 245–375]; [J.H. Rubinstein, Proc. Georgia Topology Conference, 1995, pp. 1–20]. This gives generalizations of theorems of Scharlemann, Thompson, Rubinstein, and Stocking. In the final section, we use this machinery to produce an algorithm to determine the bridge number of a knot, provided thin position for the knot coincides with bridge position. We also present several algorithmic and finiteness results about Dehn fillings with small Heegaard genus.