Incompressible Surface Research Articles

The Hilbert–Smith conjecture for three-manifolds

We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of $\mathbb Z_p$ (the $p$-adic integers) on a connected three-manifold. If $\mathbb Z_p$ acts faithfully on $M^3$, we find an interesting $\mathbb Z_p$-invariant open set $U\subseteq M$ with $H_2(U)=\mathbb Z$ and analyze the incompressible surfaces in $U$ representing a generator of $H_2(U)$. It turns out that there must be one such incompressible surface, say $F$, whose isotopy class is fixed by $\mathbb Z_p$. An analysis of the resulting homomorphism $\mathbb Z_p\to\operatorname{MCG}(F)$ gives the desired contradiction. The approach is local on $M$.

The Kakimizu complex of a connected sum of links

We show that | M S ( L 1 # L 2 ) | = | M S ( L 1 ) | × | M S ( L 2 ) | × R |\mathrm {MS}(L_1\# L_2)|=|\mathrm {MS}(L_1)|\times |\mathrm {MS}(L_2)|\times \mathbb {R} when L 1 L_1 and L 2 L_2 are any non-split and non-fibred links. Here M S ( L ) \mathrm {MS}(L) denotes the Kakimizu complex of a link L L , which records the taut Seifert surfaces for L L . We also show that the analogous result holds if we study incompressible Seifert surfaces instead of taut ones.

On knots with icon surfaces

An ICON surface is an incompressible compact orientable nonseparating surface properly embedded in a knot exterior. We show that for any odd positive number n, there exist plenty of knots whose exteriors E contain an ICON surface F with F n. We also show that our examples satisfy the -conjecture, that is, 1(E F) .

Three-dimensional transient complex free surface flows: Numerical simulation of XPP fluid

In this paper we present a finite difference MAC-type approach for solving three-dimensional viscoelastic incompressible free surface flows governed by the eXtended Pom–Pom (XPP) model, considering a wide range of parameters. The numerical formulation presented in this work is an extension to three-dimensions of our implicit technique [Journal of Non-Newtonian Fluid Mechanics 166 (2011) 165–179] for solving two-dimensional viscoelastic free surface flows. To enhance the stability of the numerical method, we employ a combination of the projection method with an implicit technique for treating the pressure on the free surfaces. The differential constitutive equation of the fluid is solved using a second-order Runge–Kutta scheme. The numerical technique is validated by performing a mesh refinement study on a pipe flow, and the numerical results presented include the simulation of two complex viscoelastic free surface flows: extrudate-swell problem and jet buckling phenomenon.

How coatings with hydrophobic particles may change the drying of water droplets: incompressible surface versus porous media effects

There is no clear statement on the role of particles in the drying of liquid marbles, which are liquid drops coated with hydrophobic solid particles. While some works report a similar drying time for liquid marbles and bare water drops others observe a faster evaporation of either liquid marbles or of bare water drops. To provide insight into the subject, we report water drying experiments in different configurations. We first focus on the drying of flat water surfaces coated with a single or several layers of hydrophobic micronic particles. Quite surprisingly, surfaces coated with a single layer of densely packed particles dry at the same speed as the bare surfaces. However, when coated with several layers of particles, the drying rate per unit surface area is significantly diminished. This effect is quantitatively explained by considering vapor diffusion through the porous media formed by the stacking of micronic particles above the interface. Then, we consider the drying of curved interfaces which are liquid marbles, i.e. drops coated with one monolayer of micronic particles. Those systematically dry faster than pure drops of the same initial volume. As the presence of a single layer of particles does not significantly affect the drying rate, this “speed-up” effect is attributed to the conservation of the surface area of the coated drop during the drying. Our quantitative experiments and understanding of the drying of liquid marbles therefore support the different results found in the literature: liquid marbles coated with one monolayer of fine solid particles do dry faster than water drops, while those coated with several layers – that may be formed by aggregates of nanoparticles – experience slower drying.

Additivity of handle number and Morse–Novikov number of a-small knots

A knot is an a-small knot if its exterior does not contain closed incompressible surfaces disjoint from some incompressible Seifert surface for the knot. Using circular thin position for knots we prove that the handle number is additive under the connected sum of two a-small knots. As a consequence the Morse–Novikov number turns out to be additive under the connected sum of two a-small knots.

Critical Heegaard surfaces obtained by amalgamation

Critical surfaces are defined by Bachman as topological index two surfaces, which are a natural extension of incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a special case, we obtain critical Heegaard surfaces by boundary stabilization. Also we give examples of unstabilized critical Heegaard surfaces—the standard Heegaard surfaces of (closed orientable surface)×S1.

LIFTED HEEGAARD SURFACES AND VIRTUALLY HAKEN MANIFOLDS

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [A. Casson and C. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283] and extends [J. Masters, W. Menasco and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004) 133–150].

INCOMPRESSIBLE ONE-SIDED SURFACES IN EVEN FILLINGS OF FIGURE 8 KNOT SPACE

In the closed, non-Haken, hyperbolic class of examples generated by (2p, q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio [Formula: see text] and determined to be unique in all cases. When applied to one-sided Heegaard splitting, this can be used to classify all geometrically incompressible splittings in this class of closed, hyperbolic examples; no analogous classification exists for two-sided Heegaard splitting.

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.

A complex of incompressible surfaces for handlebodies and the mapping class group

For a genus g handlebody H g a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H g , is constructed and several properties are established. In particular, this complex naturally contains, as a subcomplex, the complex of curves of the surface \({\partial H_{g}}\). As in the classical theory, the group of automorphisms of this complex is identified with the mapping class group of the handlebody.

On the Neuwirth conjecture for knots

Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this question and prove it for all knots up to 11 crossings except for two examples. We also establish the conjecture for all Montesinos knots and for all generalized arborescently alternating knots. For knot exteriors containing closed incompressible surfaces satisfying a simple homological condition, we establish that the knots satisfy the Neuwirth conjecture. If there is a proper degree one map from knot $K$ to knot $K'$ and $K'$ satisfies the Neuwirth conjecture then we prove the same is true for knot $K$. Algorithms are given to decide if a knot satisfies the various versions of the Neuwirth conjecture and also the related conjectures about whether all non-trivial knots have essential surfaces at integer boundary slopes.

Homology Curve Complex

A homological analogue of curve complex of a closed connected orientable surface is developed and studied. The dis-tance in this complex is shown to be quite computable and an algorithm given (Theorem 3.5). As an application of this complex it is shown that for a closed orientable 3-manifold, and any of its Heegaard splittings, one can give an algorithm to decide whether the manifold contains a 2-sided, non-separating, closed incompressible surface (Theorem 1.1).

Minimal surfaces in quasi-Fuchsian 3-manifolds

In this paper, we prove that if a quasi-Fuchsian 3-manifold M contains a closed geodesic with complex length \({\fancyscript {L} = l + i\theta}\) such that \({|\theta|/l \gg 1}\) , where l > 0 and −π ≤ θ < π, then it contains at least two incompressible minimal surfaces near the geodesic.

A new particle method for simulation of incompressible free surface flow problems

SUMMARYA new Lagrangian particle method called the consistent particle method (CPM), which solves the Navier–Stokes equations in a semi‐implicit time stepping scheme, is proposed in this paper. Instead of using kernel function as in some particle methods, partial differential operators are approximated in a way consistent with Taylor series expansion. A boundary particle recognition method is applied to help define the changing liquid domain. The incompressibility condition of free surface particles is enforced by an adjustment scheme. With these improvements, the CPM is shown to be robust and accurate in long time simulation of free surface flow particularly for smooth pressure solution. Two types of free surface flow problems are presented to verify the CPM, that is, two‐dimensional dam break and liquid sloshing in a rectangular tank. In the dam break example, the CPM solutions of pressure and wave elevation are in good agreement with published experimental results. In addition, an experimental study of water sloshing in tank on a shake table was conducted to verify the CPM solutions. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Numerical simulation for the free surface flow around a complex ship hull form at different Froude numbers

The incompressible turbulent free surface flow around the complex hull form of the DTMB 5415 model at two different speeds has been numerically simulated using the RANSE code CFX. The Volume of Fluid method (VOF) has been used with CFX for capturing the free surface flow around the ship model at the two speeds. The simulation conditions are the ones for which experimental and numerical results exist. The standard k–ɛ turbulence model has been used in CFX code. The grid generator ICEM CFD has been used for building the hybrid grid for the RANSE code solver. The results compare well with the available experimental and numerical data.

Generalizations of Agol’s inequality and nonexistence of tight laminations

We give a general lower bound for the normal Gromov norm of genuine laminations in terms of the topology of the complementary regions. In the special case of 3-manifolds, this yields a generalization of Agol's inequality from incompressible surfaces to tight laminations. In particular, the inequality excludes the existence of tight laminations with nonempty guts on 3-manifolds of small simplicial volume.

Incompressible pairwise incompressible surfaces in link complements

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S 3 – L. We discuss the properties that the surface F intersects with 2-spheres in S 3 – L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves ( R-move and S 2-move), and define the characteristic number of the topological graph for F ∩ S ± 2 . The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating (or almost alternating) link complements. We prove that the genus of the surface equals zero if the component number of F ∩ S + 2 (or F ∩ S − 2 ) is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if # ( ∂ F ) ≤ 8 .

On the mapping class group of a genus 2 handlebody

A complex of incompressible surfaces in a handlebody is constructed so that it contains, as a subcomplex, the complex of curves of the boundary of the handlebody. For genus 2 handlebodies, the group of automorphisms of this complex is used to characterize the mapping class group of the handlebody. In particular, it is shown that all automorphisms of the complex of incompressible surfaces are geometric, that is, induced by a homeomorphism of the handlebody.

Velocity field and pressure distribution around a collapsing cavitation bubble during necking and splitting

The hydrodynamic behavior of the fluid around a cavitation bubble located above a rigid boundary is investigated numerically. The liquid around the cavitation bubble is assumed to be incompressible, inviscid and irrotatational and surface tension is assumed to be negligible. Boundary-integral-equation and finite-difference methods are employed to study the problem. Three cases are investigated: (1) when the Bjerknes force is negligible in comparison with the buoyancy force, (2) when the buoyancy force is negligible in comparison with the Bjerknes force, (3) when the Bjerknes attraction force through the rigid surface and the buoyancy force are comparable. It is shown that during the collapse phase in the third case, an annular liquid jet develops around the bubble, causing it to take the shape of an hour-glass. This phenomenon is called necking which is followed by splitting the bubble into two parts. Features to note are the large lateral pressures and the high relative velocities of the fluid particles near the annular liquid jet of the bubble. This large lateral pressure may be the cause of bubble collapse. The velocity field of the liquid domain around the two parts of the cavitation bubble after splitting shows that a stagnation point exists between the two parts of the bubble. Because of the unsteady nature of the problem, the stagnation point and the point of maximum pressure do not coincide.