Closed incompressible surfaces in knot complements

In this paper we show that given a knot or link in a -plat projection with and , where is the length of the plat, if the twist coefficients all satisfy then has at least nonisotopic essential meridional planar surfaces. In particular if is a knot then contains closed incompressible surfaces. In this case the closed surfaces remain incompressible after all surgeries except perhaps along a ray of surgery coefficients in .

     

The Turaev genus of an adequate knot

The Turaev genus of a knot is an obstruction to the knot being alternating. An adequate knot is a generalization of an alternating knot. A natural problem is a characterization of the Turaev genus of an adequate knot. In this paper, we show that the Turaev genus of an adequate knot is realized by the genus of the Turaev surface associated to an adequate diagram of the knot using the Khovanov homology. As a result, we obtain the additivity of the Turaev genus of adequate knots, and show that the Turaev genus of an adequate knot is “often” preserved under mutation. We also show that an n-semi-alternating knot is of Turaev genus n. This is the first examples of adequate knots of Turaev genus two or more.     

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.

     

Incompressible pairwise incompressible surfaces in almost alternating knot complements

In this paper, we deal with incompressible pairwise incompressible surfaces in almost alternating knot complements. We show that the genus of a surface in an almost alternating knot exterior equals zero if there are two, four or six boundary components in the surface.     

Free surface motion of an incompressible ideal fluid

We consider the free boundary problem for an incompressible ideal fluid in the two-dimensional space. We show the unique existence of the solution, locally in time, even if the initial surface and the bottom are uneven.     

Closed incompressible surfaces of genus two in 3-bridge knot complements

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.     

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.     

The computational complexity of knot genus and spanning area

We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant is NP-hard.

     

Oscillation of membranes in an incompressible fluid

This article considers the problem of natural oscillations of membranes in an unbounded incompressible fluid. The article contains a qualitative analysis of the equations used and upper and lower bounds are obtained for the natural frequencies of the membrane. Translated fromDinamicheskic Sistemy. Vol. 12. pp. 23–29, 1993.     

On the existence of free surface problem for viscous incompressible flow

In this paper we are concerned with the flow of a viscous, incompressible fluid in a bounded, three-dimensional region Ω with free surface boundary conditions. Using a method introduced by the author, that consider a two-fluid system in which the atmosphere or the vacuum is considered as a second fluid, separated from the first one by a free interface Γ(t), we prove existence of a kind of weak solution that we call quasi-weak solution.

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Sunto In questo lavoro studiamo il moto di un fluido viscoso e incomprimibile in una regione limitata tridimensionale Ω, con condizioni al contorno di superficie libera. Utilizzando un metodo, dovuto all’autore, che consiste nel considerare l’atmosfera o il vuoto come un secondo fluido, separato dal primo da un’interfaccia mobile Γ(t), dimostriamo l’esistenza di una sorta di soluzione debole, denominata soluzione quasi-debole.

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Work supported by Progetto Murst n. 9801262841.     

On an incompressible model in radiation hydrodynamics

We consider a simplified model arising in radiation hydrodynamics based on the incompressible Navier–Stokes–Fourier system describing a macroscopic fluid motion coupled to a transport equation modeling the propagation of radiative intensity. We establish global‐in‐time existence for the associated initial‐boundary value problem in the framework of weak solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

A partial differential equation which describes an interatomic surface

In this work we present a first-order partial differential equationwhich defines the topology of single ‘atomic entities’in multiatomic systems. Such an equation, obtained by R. F.W. Bader, is here analysed and discussed from a general mathematicalpoint of view; a method is then proposed for defining the initialor boundary condition. With this contribution we would liketo promote and stimulate a more detailed analysis which goesbeyond practical purposes and basic mathematical analysis inorder to have a deeper understanding of the theory behind theequation and its consequences for practical applications.     

On zeros of the Alexander polynomial of an alternating knot

We prove that for any zero α of the Alexander polynomial of a two-bridge knot, −3<Re(α)<6. Furthermore, for a large class of two-bridge knots we prove −1<Re(α).     

Steady magnetohydrodynamic flow of an incompressible viscous fluid involving an ignorable co-ordinate

Résumé On considère l'écoulement visqueux régulier et incompressible ainsi que la distribution de température d'un fluide conducteur d'électricité contenu dans un conduit. En utilisant une forme spéciale du potentiel vectoriel magnétique et en changeant les variables des coordonnées, on démontre la possibilité de réduire le problème de l'écoulement à la résolution de l'équation de Poisson et de deux équations indépendantes linéaires différentielles et partielles du deuxième ordre. Les résultats obtenus sont employés pour déterminer l'écoulement et la distribution de la température d'une forme peu commune de l'écoulement de Couette.     

Non-linear near-resonance oscillations of an elastic incompressible layer

Plane one-dimensional waves of small amplitude, propagating transverse to an incompressible elastic layer and reflected successively from its boundaries, are considered. The oscillations are caused by small periodic (or close to periodic) external action on one of the layer boundaries, when the period of the external action is close to the period of natural oscillations of the layer. One of the boundaries of the elastic layer is fixed, while the other performs small specified two-dimensional motion in its plane. In such a near-resonance situation, non-linear effects occur which may build up over time. A system of equations is obtained which describes the slow change in the functions characterizing the oscillations of the medium in each period of the external action. It is assumed that all the quantities depend both on real time, any change of which in the approach considered is limited to one period, and on “slow” time, for which one period of real time serves as a small quantity. It is assumed that the evolution of the solution occurs when the slow time changes, while the role of real time is similar to the role of a spatial variable. This system of equations is obtained by the method of averaging over a period of the quantities representing nonlinear terms and the effect of the boundary conditions in the equations. It contains derivatives with respect to the real and slow times and also values of the functions characterizing the solution averaged over a period of the real time. If the averaged values are known, the equations have a hyperbolic form and their solutions can be both continuous and contain weak and strong discontinuities.