Incompressible Surface Research Articles

Separating incompressible surfaces and stabilizations of Heegaard splittings

We describe probably the simplest 3-manifold which contains closed separating incompressible surfaces of arbitrarily large genus. Two applications of this observation are given. (1) For any closed, orientable 3-manifold .

Contact structures on elliptic $3$-manifolds

We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure if and only if the corresponding group of deck transformations on S 3 (after possibly conjugating by an isometry) preserves the standard contact structure. We also relate universally tight contact structures on 3-manifolds covered by S 3 to the isomorphism SO(4) = (SU(2) ◊ SU(2))/±1. The main tool used is equivariant framings of 3-manifolds. A contact structure on a 3-dimensional manifold M is a smooth, totally non- integrable tangent plane field, i.e., a tangent plane field locally of the form = ker( ) for a 1-form such that ^ d is everywhere non-degenerate. We shall assume that M is oriented. We say is positive if the orientation on M agrees with that induced by the volume form ^ d . Observe that the orientation of ^d does not depend on the sign of , and is thus determined by (even though = ker( ) only locally). A central role in understanding 3-dimensional manifolds has been played by co- dimension one structures - surfaces, foliations and laminations - in these manifolds. Without additional conditions such structures always exist, and are of not much consequence. However, the presence of essential co-dimension one structures - incompressible surfaces, taut foliations and essential laminations, leads to deep

KAUFFMAN–HARARY CONJECTURE HOLDS FOR MONTESINOS KNOTS

The Kauffman–Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize this conjecture by stating it in terms of homology of the double cover of S3. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman–Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman–Harary conjecture for all Montesinos links. Finally, we discuss on the relation between the conjecture and Menasco's work on incompressible surfaces in exteriors of alternating links.

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

AbstractThis paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.

Small 3-Manifolds of Large Genus

We prove the existence of pure braids with arbitrarily many strands which are small, i.e. the complements contain no closed incompressible surface which is not boundary parallel. This implies the existence of irreducible non-Haken 3-manifolds of arbitrarily high Heegaard genus.

MERIDIONALLY INCOMPRESSIBLE SURFACES AND THE DISTANCE OF (1,1)-SPLITTINGS

In this paper, we show that for any torus giving a (1,1)-splitting of (M,K), every closed orientable surface of genus g>0 which is meridionally incompressible can be isotoped to intersect the the splitting torus in a circle. As an application, we estimate the distance, which is defined in [16], of (1,1)-splittings of (1,1)-knots whose exteriors contain closed orientable meridionally incompressible surfaces.

Computation of free surface flows with a projection FEM in a moving mesh framework

A numerical model for the simulation of incompressible viscous free surface flows is presented. The equations governing the fluid flow are solved with the finite element fractional-step projection method. A moving mesh approach, similar to the arbitrary Lagrangian–Eulerian method is introduced to deal with the presence of a free surface. Validations are carried out with a two-dimensional sloshing problem with small deformations. Thus, results concerning, respectively, the damped oscillations of a gravity and a capillary wave are compared with the initial-value linear theory. Then, a three-dimensional computation concerning the behavior of a uniform flow field subjected on its free surface to an external applied traction is presented.

Totally knotted Seifert surfaces with accidental peripherals

We show that if there exists an essential accidental surface in the knot exterior, then a closed accidental surface also exists. As its corollary, we know boundary slopes of accidental essential surfaces are integral or meridional. It is shown that an accidental incompressible Seifert surface in knot exteriors in S 3 S^3 is totally knotted. Examples of satellite knots with arbitrarily high genus Seifert surfaces with accidental peripherals are given, and a Haken 3-manifold which contains a hyperbolic knot with an accidental incompressible Seifert surface of genus one is also given.

Longitudinal slope and Dehn fillings

Let M be an irreducible 3-manifold with an incompressible torus boundary T, and g a slope on T, which bounds an incompressible surface, with genus g say. We assume that there exists a slope r that produces an essential 2-sphere by Dehn filling. Let q be the minimal geometric intersection number between the essential 2-sphere and the core of the Dehn filling. Then, we show that q 1⁄4 2 or the minimal geometric intersection number between g and r is bounded by 2g 1. In the special case that M is the exterior of a non-cable knot K in S, we show that qb 6 and jrja 2g 1, where g is the genus of the knot K. We get also similar and simpler results for the projective slopes. These imply immediately a known result that the cabling and RP conjectures are true for genus one knots.

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.)

Boundary curves of surfaces with the 4–plane property

Let M be an orientable and irreducible 3-manifold whose boundary is an incompressible torus. Suppose that M does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in M with the 4-plane property can realize only finitely many boundary slopes. Moreover, we will show that only finitely many Dehn fillings of M can yield 3-manifolds with nonpositive cubings. This gives the first examples of hyperbolic 3-manifolds that cannot admit any nonpositive cubings.

Three‐dimensional numerical modelling of free surface flows with non‐hydrostatic pressure

AbstractA three‐dimensional numerical model is developed for incompressible free surface flows. The model is based on the unsteady Reynolds‐averaged Navier–Stokes equations with a non‐hydrostatic pressure distribution being incorporated in the model. The governing equations are solved in the conventional sigma co‐ordinate system, with a semi‐implicit time discretization. A fractional step method is used to enable the pressure to be decomposed into its hydrostatic and hydrodynamic components. At every time step one five‐diagonal system of equations is solved to compute the water elevations and then the hydrodynamic pressure is determined from a pressure Poisson equation. The model is applied to three examples to simulate unsteady free surface flows where non‐hydrostatic pressures have a considerable effect on the velocity field. Emphasis is focused on applying the model to wave problems. Two of the examples are about modelling small amplitude waves where the hydrostatic approximation and long wave theory are not valid. The other example is the wind‐induced circulation in a closed basin. The numerical solutions are compared with the available analytical solutions for small amplitude wave theory and very good agreement is obtained. Copyright © 2002 John Wiley & Sons, Ltd.

Closed incompressible surfaces in the complements of positive knots

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splittability and the primeness of positive knots and links can be seen from their positive diagrams.

Critical Heegaard surfaces

In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.

Heegaard splittings of Haken manifolds have bounded distance

Given a Heegaard splitting and an incompressible surface S and a Heegaard splitting of an irreducible manifold, I shall use a generalization of Haken's lemma proved by Kobayashi in order to define a pair of simple closed curves on the splitting surface such that each bounds a disc in one of the handlebodies of the splitting. By modifying the proof of Kobayashi's lemma, I shall show that the sequence of boundary compressions used to isotope S places a bound on the distance between these two simple closed curves in the complex of curves. This will then place a bound on the distance of the Heegaard splitting.

On the existence of infinitely many essential surfaces of bounded genus

A theorem of William Jaco and Eric Sedgwick states that if M is an irreducible, ∂-irreducible 3-manifold with boundary a single torus, and if M contains no genus one essential (incompressible and ∂-incompressible) surfaces, then M cannot contain infinitely many distinct isotopy classes of essential surfaces of uniformly bounded genus. The main result in this paper is a generalization: If M is an irreducible ∂-irreducible 3-manifold with boundary, and M contains no genus one or genus zero essential surfaces, then M cannot contain infinitely many isotopy classes of essential surfaces of uniformly bounded genus.

Numerical simulation of the draft tube and tailwater flow interaction

This work focuses on the numerical simulation of the interaction between the draft tube and tailwater flow in low head hydro power plants. An approach to the solution of incompressible free surface flow problems is developed. The flow field and the free surface location are calculated by coupling the free surface kinematic and dynamic conditions with the equations of motion for the bulk flow. A pressure correction method is applied to the flow calculation in a moving grid system. The RNG k-ε model is used for turbulence modelling. In order to reduce the computational time, the PVM (Parallel Virtual Machine) subroutines are implemented.

Genus two 3–manifolds are built from handle number one pieces

Let M be a closed, irreducible, genus two 3-manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M. Then each component manifold M_i of M-F has handle number at most one, i.e. admits a Heegaard splitting obtained by attaching a single 1-handle to one or two components of boundary M_i. This result also holds for a decomposition of M along a maximal collection of incompressible tori.

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.

Pre-taut sutured manifolds and essential laminations

In 1989, D. Gabai and U. Oertel [8] introduced the concept of the essential lamination, which is a hybrid object lying between incompressible surfaces and taut foliations, and generalizing both. We say that a 3-manifold is laminar if it contains an essential lamination. An important result of [8] is that the universal covers of laminar manifolds are homeomorphic to R. This fact furnishes a strong method for studying the manifolds obtained by Dehn surgery along knots, especially concerning Property P Conjecture (nontrivial Dehn surgery on a nontrivial knot in 3 never yields a simply-connected manifold) and Cabling Conjecture (Dehn surgery on a non-cable knot cannot yield a reducible manifold). For example, see [4] for non-torus alternating knots, [3], [12] for 2-bridge knots, [17] for most algebraic knots and [9] for knots with some kind of essential tangle decompositions. We note that by [8] a 3-manifold is laminar if and only if it contains an essential branched surface (for the definition see §2), and the above authors who followed [8] obtained their results by constructing essential branched surfaces. We note that sutured manifold theory was used in [14] and [18]. One of their approaches is to construct a closed essential branched surface in the exterior ( ) of a knot and show that remains essential after any nontrivial Dehn filling along ∂ ( ) (we call such persistently essential). Then we see, by [8], that has Property P in a strong form and that the cabling conjecture is true for . (We say that a knot has strong Property P if every manifold obtained by a nontrivial Dehn surgery along has universal cover R.) It is, however, an open question whether or not every knot with strong Property P admits a persistently essential lamination in its complement. In [1], [2], M. Brittenham had a paradigm shift in proving strong Property P for knots. Instead of constructing a branched surface in the complement of a given knot, he first constructed a branched surface and then embedded a knot in its complement. More precisely, he first constructed a closed branched surface in 3 from any in-