A formula for the braid index of links

Morton and Franks–Williams independently gave a lower bound for the braid index b(L) of a link L in S³ in terms of the v-span of the Homfly-pt polynomial P_L(v,z) of L: . Up to now, many classes of knots and links satisfying the equality of this Morton–Franks–Williams's inequality have been founded. In this paper, we give a new such a class of knots and links and make an explicit formula for determining the braid index of knots and links that belong to the class . This gives simultaneously a new class of knots and links satisfying the Jones conjecture which says that the algebraic crossing number in a minimal braid representation is a link invariant. We also give an algorithm to find a minimal braid representative for a given knot or link in .     

The Space of subcontinua of a 2-dimensional continuum is infinite dimensional

Let be a metric continuum and let denote the space of subcontinua of with the Hausdorff metric. We settle a longstanding problem showing that if then . The special structure and properties of hereditarily indecomposable continua are applied in the proof.

     

A faithfulness criterion for the Gassner representation of the pure braid group

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, . Long and Paton proved that if a Burau matrix has ones on the diagonal and zeros below the diagonal then is the identity matrix. In this paper, a generalization of Long and Paton's result will be proved. Our main theorem is that if the trace of the image of an element of under the reduced Gassner representation is , then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.

     

Specialized Runge-Kutta methods for index differential-algebraic equations

We consider the numerical solution of systems of semi-explicit index differential-algebraic equations (DAEs) by methods based on Runge-Kutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK methods have been proposed. In this paper we propose a simple alternative which does not require any extra projection step and does not use any additional internal stage. Moreover, symmetry of Gauss methods is preserved. The main idea is to replace the satisfaction of the constraints at the internal stages in the standard definition by enforcing specific linear combinations of the constraints at the numerical solution and at the internal stages to vanish. We call these methods specialized Runge-Kutta methods for index DAEs (SRK-DAE).

     

The fundamental group of the complement of the branch curve of the second Hirzebruch surface

In this paper we prove that the Hirzebruch surface F_2,(2,2) embedded in supports the conjecture on the structure and properties of fundamental groups of complement of branch curves of generic projections, as laid out in [M. Teicher, New Invariants for surfaces, Contemp. Math. 231 (1999) 271–281]. We use the regeneration from [M. Friedman, M. Teicher, The regeneration of a 5-point, Pure and Applied Mathematics Quarterly 4 (2) (2008) 383–425. Fedor Bogomolov special issue, part I], the van Kampen theorem and properties of -groups [M. Teicher, On the quotient of the braid group by commutators of transversal half-twists and its group actions, Topology Appl. 78 (1997) 153–186], where is a quotient of the braid group B_n, for n=16.     

Some sufficient conditions for tunnel numbers of connected sum of two knots not to go down

In this paper, we show the following result: Let K _i be a knot in a closed orientable 3-manifold M _i such that (M _i ,K _i ) is not homeomorphic to (S ₂ ×S ₁, x ₀ ×S ₁), i = 1, 2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E(K _i ) is less than the difference of one and twice of the tunnel number of K _i . Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive.     

Non-additivity for triple point numbers on the connected sum of surface-knots

Any surface-knot in 4-space can be projected into 3-space with a finite number of triple points, and its triple point number, , is defined similarly to the crossing number of a classical knot. By definition, we have for the connected sum. In this paper, we give infinitely many pairs of surface-knots for which this equality does not hold.

     

Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$-actions and the locus of Gorenstein stable surfaces

In this note, the geography of minimal surfaces of general type admitting ${\mathbb{Z}}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ contains surfaces admitting a ${\mathbb{Z}}_2^2$-action for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$ or $K^2=8\chi$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification ${\overline{\mathfrak{M}}}_{K^2,\chi }$ of Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ for every admissible pair $(K^2,\chi )$ such that $2\chi -6\le K^2\le 8\chi -8$.     

The class of groups which have a subgroup of index 2 is not elementary

F. Oger proved that if A is a finite group, then the class of groups which are abelian-by-A can be axiomatized by a single first order sentence. It is established here that, in Oger's result, the word abelian cannot be replaced by group. Received: 15 March 1996 / Published online: 18 July 2001     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.

     

On product-cordial index sets and friendly index sets of 2-regular graphs and generalized wheels

A vertex labeling f : V → Z2 of a simple graph G = (V, E) induces two edge labelings f+ , f*: E → Z2 defined by f+ (uv) = f(u)+f(v) and f*(uv) = f(u)f(v). For each i∈Z2 , let vf(i) = |{v ∈ V : f(v) = i}|, e+f(i) = |{e ∈ E : f+(e) = i}| and e*f(i)=|{e∈E:f*(e)=i}|. We call f friendly if |vf(0)-vf(1)|≤ 1. The friendly index set and the product-cordial index set of G are defined as the sets{|e+f(0)-e+f(1)|:f is friendly} and {|e*f(0)-e*f(1)| : f is friendly}. In this paper we study and determine the connection between the friendly index sets and product-cordial index sets of 2-regular graphs and generalized wheel graphs.     

A mod 2 index theorem for the twisted Signature operator

A mod 2 index theorem for the twisted Signature operator on 4 _q + 1 dimensional manifolds is established. This result generalizes a result of Farber and Turaev, which was proved for the case of orthogonal flat bundles, to arbitrary real vector bundles. It also provides an analytic interpretation of the sign of the Poincare-Reidemeister scalar product defined by Farber and Turaev. Project partially supported by the National Natural Science Foundation of China (Grant No. 19525102), the Fok Ying-Tung Foundation and the Qiu Shi Foundation.     

补图为2-点或2-边连通的图的最小特征值

图的最小特征值定义为图的邻接矩阵的最小特征值,是刻画图结构性质的一个重要代数参数. 在所有给定阶数的补图为2-点或2-边连通的图中, 刻画了最小特征值达到极小的唯一图, 并给出了这类图最小特征值的下界.     

The hamiltonian index of a 2-connected graph

Let G be a graph. Then the hamiltonian index h(G) of G is the smallest number of iterations of line graph operator that yield a hamiltonian graph. In this paper we show that for every 2-connected simple graph G that is not isomorphic to the graph obtained from a dipole with three parallel edges by replacing every edge by a path of length l≥3. We also show that for any two 2-connected nonhamiltonian graphs G and with at least 74 vertices. The upper bounds are all sharp.     

Use of the Szeged index and the revised Szeged index for measuring network bipartivity

We have revisited the Szeged index (Sz) and the revised Szeged index (Sz^∗), both of which represent a generalization of the Wiener number to cyclic structures. Unexpectedly we found that the quotient of the two indices offers a novel measure for characterization of the degree of bipartivity of networks, that is, offers a measure of the departure of a network, or a graph, from bipartite networks or bipartite graphs, respectively. This is because the two indices assume the same values for bipartite graphs and different values for non-bipartite graphs. We have proposed therefore the quotient Sz/Sz^∗ as a measure of bipartivity. In this note we report on some properties of the revised Szeged index and the quotient Sz/Sz^∗ illustrated on a number of smaller graphs as models of networks.     

`CH~2中齐性曲面的分类(英文)

ＣＨ~２中的曲面 Ｍ称为（局部）齐性的,如果对任何一对点 ｐ,ｑ,存在变换。 σ　Ｕ（１,２）把含ｐ的一个邻域变为含ｑ的邻域,且把ｑ变为ｑ．本文给出ＣＨ２中齐性曲面的完全分类．