Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links

Let F be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle (B,T). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B,T) and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead to an algebraic structure which is isomorphic to the rational numbers.We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link K. In particular we show that if K is a knot, then the complement of K does not contain such a surface.     

Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

Kirchhoff systems with dynamic boundary conditions

We are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the p-Kirchhoff operator, under dynamic boundary conditions, when p>p_n with p_n<2. The systems involve nonlinear external forces and may be affected by a perturbation of the type |u|^p−2u. Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the source term f and the external nonlinear boundary damping Q.     

Pre-Bloch invariants of 3-manifolds with boundary

Let M be an oriented hyperbolic 3-manifold with finite volume. In [W.D. Neumann, J. Yang, Bloch invariants of hyperbolic 3-manifolds, Duke Math. J. 96 (1999) 29-59. [9]], Neumann and Yang defined an element β(M) of Bloch group B(C) for M. For this β(M), volume and Chern-Simons invariant of M is represented by a transcendental function. In this paper, we define β(M,ρ,C,o)∈P(C) for an oriented 3-manifold M with boundary, a representation of its fundamental group , a pants decomposition C of ∂M and an orientation o on simple closed curves of C. Unlike in the case of finite volume, we construct an element of pre-Bloch group P(C), and we need essentially the pants decomposition on the boundary. The volume makes sense for β(M,ρ,C,o) and we can describe the variation of volume on the deformation space.     

On a surface singular braid monoid

We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follow from topological Yoshikawa moves. As a direct application we will reprove some known theorem about twist-spun knots. We wish then to investigate an index associated to the closure of surface singular braid. Using our relations we will prove that there are exactly six types of knotted surfaces with the index less or equal to two, and there are infinitely many types of surface-knots with index equal to three. Towards the end we will construct a family of classical diagrams such that to unlink them requires at least four Reidemeister III moves.     

Any chart with at most one crossing is a ribbon chart

In this paper, we shall show the one written in the title.     

Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo–Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fréchet space. Besides, we use compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorohod theorem in non-metric spaces. The general framework is applied to the stochastic Navier–Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.     

Sources and sinks for Axiom A flows on closed 3-manifolds

We prove that an Axiom A vector field on an orientable closed 3-manifold not homeomorphic toS³ for which every transverse torus bounds a solid torus either is transitive or has a sink or a source. This result is false without these hypotheses.     

Hyperbolic spatial graphs in 3-manifolds

For any closed connected orientable 3-manifold M, we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S³ via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M.     

A computer search for contractible 3-manifolds

A theorem is proved giving a necessary condition for a standard spine of a prime homology 3-ball to be minimal. This result enables an inspection of all standard spines with five or fewer vertices, by computer. There are exactly two such minimal spines, one for the 3-ball and one for the dodecahedral homology-ball of Poincaré. Corollary: No counterexample to the Poincaré Conjecture exists having a standard spine with five or fewer vertices.     

The structure of 3-manifolds with two-generated fundamental group

We provide a structure theorem for 3-manifolds with 2-generated fundamental group and non-trivial JSJ-decomposition. We further give a number of applications.     

An effective algorithm to get linking numbers of 3-manifolds

Summary A colored triangulation of a 3-manifoldM ³ is a decomposition into tetrahedra so that each vertex of them receive one of the colors 0, 1, 2, 3 in such a way that each tetrahedron has four differently colored vertices. From the combinatorics of the dual of a colored triangulation forM ³ we provide an easy algorithm to get a special kind of intersection matrix; from this matrix and from the torsion coefficients of the firstZ-homology group ofM ³ we provide a formula which yields its linking numbers.     

Branched spines and contact structures on 3-manifolds

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in a given homotopy class of plane fields. Entrata in Redazione il 6 novembre 1998.     

Embeddability of multiple cones

The main result of this paper is that if X is a Peano continuum such that its nth cone Cn(X) embeds into Rn+2 then X embeds into S². This solves a problem proposed by W. Rosicki.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.     

Attractors for non-dissipative irrotational von Karman plates with boundary damping

Long-time behavior of solutions to a von Karman plate equation is considered. The system has an unrestricted first-order perturbation and a nonlinear damping acting through free boundary conditions only.This model differs from those previously considered (e.g. in the extensive treatise (Chueshov and Lasiecka, 2010 [11])) because the semi-flow may be of a non-gradient type: the unique continuation property is not known to hold, and there is no strict Lyapunov function on the natural finite-energy space. Consequently, global bounds on the energy, let alone the existence of an absorbing ball, cannot be a priori inferred. Moreover, the free boundary conditions are not recognized by weak solutions and some helpful estimates available for clamped, hinged or simply-supported plates cannot be invoked.It is shown that this non-monotone flow can converge to a global compact attractor with the help of viscous boundary damping and appropriately structured restoring forces acting only on the boundary or its collar.     

A disjoint disks property for 3-manifolds

We show that the map separation property (MSP), a concept due to H.W. Lambert and R.B. Sher, is an appropriate analogue of J.W. Cannon’s disjoint disks property (DDP) for the class $\text{C}$ of compact generalized 3-manifolds with zero-dimensional singular set, modulo the Poincaré conjecture. Our main result is that the Poincaré conjecture (in dimension three) is equivalent to the conjecture that every X?$\text{C}$ with the MSP is a topological 3-manifold.     

Incompressible surfaces in handlebodies and boundary reducible 3-manifolds

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.     

On the existence of representations of finitely presented groups in compact Lie groups

Given a finite, connected 2-complex X   such that _b2(X)?1$b_2(X)?1$ we establish two existence results for representations of the fundamental group of X into compact connected Lie groups G  , with prescribed values on certain loops. If _b2(X)=1$b_2(X)=1$ we assume G=SO(3)$G=\mathrm{SO}(3)$ and that the cup product on H¹(X;Q)$H^1(X;\mathbb{Q})$ is non-zero.