A chain complex and Quadrilaterals for normal surfaces

The Unknotting Problem and Normal Surface Q-Theory

Tollefson described a variant of normal surface theory for 3-manifolds, called Q-theory, where only the quadrilateral coordinates are used. Suppose $M$ is a triangulated, compact, irreducible, boundary-irreducible 3-manifold. In Q-theory, if $M$ contains an essential surface, then the projective solution space has an essential surface at a vertex. One interesting situation not covered by this theorem is when $M$ is boundary reducible, e.g. $M$ is an unknot complement. We prove that in this case $M$ has an essential disc at a vertex of the Q-projective solution space.

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.

2-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 tubes, an octagon and a tube, or a 12-gon. In this paper we use the theory of critical surfaces developed in earlier work to prove the existence of topologically interesting 2-normal surfaces. Our main results are (1) if a ball with normal boundary in a triangulated 3-manifold contains two almost normal 2-spheres then it contains a 2-normal 2-sphere and (2) in …

Locally Helical Surfaces have bounded twisting

A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and helicoids. Helical pieces can have any number of positive or negative twists. We show here that the net twisting of the helical pieces of any such surface in a given triangulated 3-manifold is bounded.

Normal surfaces as combinatorial slicings

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a lower bound on the number of quadrilaterals of normal surfaces depending on the genus g is presented. It is shown to be sharp for infinitely many values of g. Furthermore we classify slicings of combinatorial 3-manifolds with a maximum number of edges in the slicing.

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 …

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 non-empty 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 non-integer 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 poi…

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible point…

Normal surfaces and colored Khovanov homology

We show that colored Khovanov homology detects classes of essential surfaces as a direct analogue of the slope conjectures for the colored Jones polynomial. We do this by identifying certain generators of the colored Khovanov chain complex with normal surfaces in the complement of the knot using an ideal triangulation from a diagram.

自由群图的非法正规链中的同调挠增长

Homological torsion growth in non-normal chains of graphs of free groups

本文证明了非同构于自由积的双曲群在分裂为带循环边群的自由群图时，存在具有指数同调挠增长的穷举嵌套有限指数非法正规子群序列。