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 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.

     

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.     

Boundary slopes of non-orientable Seifert surfaces for knots

We study non-orientable Seifert surfaces for knots in the 3-sphere, and examine their boundary slopes. In particular, it is shown that for a crosscap number two knot, there are at most two slopes which can be the boundary slope of its minimal genus non-orientable Seifert surface, and an infinite family of knots with two such slopes will be described. Also, we discuss the existence of essential non-orientable Seifert surfaces for knots.     

Infinitely many meridional essential surfaces of bounded genus in hyperbolic knot exteriors

Geometriae Dedicata - We show the existence of an infinite collection of hyperbolic knots where each of which has in its exterior meridional essential planar surfaces of arbitrarily large number of...     

A non symmetric FVM-BEM coupling method

We consider the prototype model for flow and transport of a concentration in porous media in an interior domain and couple it with a diffusion process in the corresponding unbounded exterior domain. To guarantee mass conservation and stability with respect to dominating convection also for a discrete solution we introduce a non symmetric coupling of the vertex-centered finite volume method (FVM) and the boundary element method (BEM). BEM approximates the unbounded exterior problem which avoids truncation of the domain. One can also interpret that the (unbounded) exterior problem “replaces” the boundary conditions of the interior problem. We aim to provide a first rigorous analysis of the discrete system for different model parameters; existence and uniqueness, convergence, and a priori estimates. Numerical examples illustrate the strength of the chosen method which is computational cheaper than the previous three field FVM-BEM couplings. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Link invariants via counting surfaces

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram. Until recently, explicit formulas of this type were known only for few invariants of low degrees. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. We then identify the resulting invariants with certain derivatives of the HOMFLYPT polynomial.     

Spanning Pre-disks in a Compression Body

A properly embedded essential planar surface P (not a disk) in a compression body V is called a spanning pre-disk with respect to J,if one boundary component of P is lying in (З)+V and all other boundary components of P are lying in (З)_ V and coplanar with J.In this paper,we show that the number of boundary components of spanning pre-disks in a compression body is unbounded.But the number of a maximal collection of spanning pre-disks is bounded.     

On the existence and stability of 2-D perturbed steady subsonic circulatory flows

In this paper, under the generalized conservation condition of mass flux in a unbounded domain, we are concerned with the global existence and stability of a perturbed subsonic circulatory flow for the two-dimensional steady Euler equation, which is assumed to be isentropic and irrotational. Such a problem can be reduced into a second order quasi-linear elliptic equation on the stream function in an exterior domain with a Dirichlet boundary value condition on the circular body and a stability condition at i...     

Once-punctured Klein bottles in knot complements

We find the family of all knots in S³ which are spanned by two essential once-punctured Klein bottles with boundary slopes at distance 4, thus settling a conjecture by K. Ichihara, M. Ohtouge, and M. Teragaito. We also address the more general question of when a knot exterior in an arbitrary 3-manifold contains two essential once-punctured Klein bottles with distinct boundary slopes.     

几乎交错纽结补中决定的不可压缩曲面

本文讨论了几乎交错纽结补中的不压缩，两两不可压缩曲面的性质．证明了，当Ｋ是素的几乎交错纽结时，若ＦＳ３－Ｋ是不可压缩、两两不可压缩曲面，则对于固定的边界分支数ｎ，曲面Ｆ的合痕类是有限的．     

The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G _D . However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F _G and S _G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.     

Crosscap numbers of pretzel knots

The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable surfaces bounded by the knot. In this paper, we determine the crosscap numbers of a large class of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos knots developed by Hatcher and Oertel.     

An Interface Problem in Solid Mechanics with a Linear Elastic and a Hyperelastic Material

The three dimensional interface problem is considered with the homogeneous Lamé system in an unbounded exterior domain and some quasistatic nonlinear elastic material behavior in a bounded interior Lipschitz domain. The nonlinear material is of the Mooney-Rivlin type of polyconvex materials. We give a weak formulation of the interface problem based on minimizing the energy, and rewrite it in terms of boundary integral operators. Then, we prove existence of solutions.     

FINITE ELEMENT NONLINEAR GALERKIN COUPLING METHOD FOR THE EXTERIOR STEADY NAVIER-STOKES PROBLEM

1.IntroductionNonlinearGalerkinmethodsaremultilevelschemesforthedissipativeevolutionpartialdifferentialequations.Theycorrespondtothesplittingsoftheunknownu:u=y z)wherethecomponentsareofdifferentorderofmagnitudewithrespecttoaparameterrelatedtothespati...     

Existence of stationary solutions of the Navier–Stokes equations in two dimensions in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here, we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

A slice genus lower bound from Khovanov–Rozansky homology

We show that a generic perturbation of the doubly-graded Khovanov–Rozansky knot homology gives rise to a lower-bound on the slice genus of a knot. We prove a theorem about obtainable presentations of surfaces embedded in 4-space, which we use to simplify significantly our algebraic computations.     

The Helmholtz equation with impedance in a half-plane

This Note gives answers to the uniqueness and existence questions for solutions of the Helmholtz equation in an half-plane with an impedance or mixed boundary condition. We deal with unbounded domains which boundaries are unbounded too. The radiation conditions are different from the ones that we found in an usual exterior problem due to the appearance of surface waves. We first compute and study the half-plane Green's function to see how the solutions behave at infinity, and second obtain integral representation for these solutions. To cite this article: M. Duran et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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.     

On α-dynamos

The function α that describes the generation of large scale magnetic fields from small scale motions is assumed to be non-zero only in a thin boundary layer. As a consequence, the problems for the poloidal and meridional fields in a symmetric rotating configuration decouple and may be solved sequentially. Analysis of the boundary layer yields equivalent boundary conditions on interior and exterior field components. The general eigenvalue problem is formulated but only the case of neutral stability is examined.