On the quantum filtration of the Khovanov-Rozansky cohomology

We prove the quantum filtration on the Khovanov-Rozansky link cohomology Hp with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to Hp that is invariant under Reidemeister moves, whose E₁ term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology Hn. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in standard contact S³.     

Writhe polynomials and shell moves for virtual knots and links

The writhe polynomial is a fundamental invariant of an oriented virtual knot. We introduce a set of local moves for oriented virtual knots called shell moves. The first aim of this paper is to prove that two oriented virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of shell moves. The second aim of this paper is to classify oriented 2-component virtual links up to shell moves by using several invariants of virtual links.     

Dimension Groups for Polynomial Odometers

Here we study a class of dynamical systems we call polynomial odometers. These are adic maps on regularly structured Bratteli diagrams and include the Pascal and Stirling adic maps as examples. We describe the dimension groups associated with these systems and use this to study spaces of invariant measures. For many, but not all, the space of invariant measures is affinely homeomorphic to the space of Borel probability measures on a closed interval in $\mathbb{R}$ , we call such polynomial odometers reasonable. We describe the possible isomorphisms between dimension groups for reasonable polynomial odometers, and use this to prove a version of a result of Giordano, Putnam and Skau for this situation. Namely, we show that there is an isomorphism between unital ordered groups associated with two reasonable polynomial odometers if and only if there is a special kind of orbit equivalence between the two.     

Link homology theories from symplectic geometry

For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapse of the bigrading. Our work is a generalization of that of Seidel and Smith, who treated the case n=2.     

Sharp estimates and a central limit theorem for the invariant law for a large star-shaped loss network

Calls arrive in a Poisson stream on a symmetric network constituted of N links of capacity C. Each call requires one channel on each of L distinct links chosen uniformly at random; if none of these links is full, the call is accepted and holds one channel per link for an exponential duration, else it is lost. The invariant law for the route occupation process has a semi-explicit expression similar to that for a Gibbs measure: it involves a combinatorial normalizing factor, the partition function, which is very difficult to evaluate. We study the large N limit while keeping the arrival rate per link fixed. We use the Laplace asymptotic method. We obtain the sharp asymptotics of the partition function, then the central limit theorem for the empirical measure of the occupancies of the links under the invariant law. We also obtain a sharp version for the large deviation principle proved in Graham and O'Connell (Ann. Appl. Probab. 10 (2000) 104).     

Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.     

Shift equivalence and the Conley index

In this paper we introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

     

Surgery obstructions from Khovanov homology

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S ³, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.     

The hexatangle

We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in S³. In particular, we want to determine when we get S³ by surgery on such a link. We consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form , where σ₁, σ₂ are the generators of the 3-braid group and e₁, f₁, e are integers. We study Dehn surgeries on these links, and determine exactly which ones admit an integral surgery producing the 3-sphere. This is equivalent to determining the surgeries of some type on a certain six component link L that produce S³. The link L is strongly invertible and its exterior double branch covers a certain configuration of arcs and spheres, which we call the hexatangle. Our problem is equivalent to determine which fillings of the spheres by integral tangles produce the trivial knot, which is what we explicitly solve. This hexatangle is a generalization of the pentangle, which is studied in [C.McA. Gordon, J. Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004) 417-485].     

Strongly regular graphs and spin models for the Kauffman polynomial

We study spin models for invariants of links as defined by Jones [22]. We consider the two algebras generated by the weight matrices of such models under ordinary or Hadamard product and establish an isomorphism between them. When these algebras coincide they form the Bose-Mesner algebra of a formally self-dual association scheme. We study the special case of strongly regular graphs, which is associated to a particularly interesting link invariant, the Kauffman polynomial [27]. This leads to a classification of spin models for the Kauffman polynomial in terms of formally self-dual strongly regular graphs with strongly regular subconstituents [7]. In particular we obtain a new model based on the Higman-Sims graph [17].     

Vassiliev invariants of type one for links in the solid torus

In Bataineh (2003) [2] we studied the type one invariants for knots in the solid torus. In this research we study the type one invariants for n-component links in the solid torus by generalizing Aicardi's invariant for knots in the solid torus to n-component links in the solid torus. We show that the generalized Aicardi's invariant is the universal type one invariant, and we show that the generalized Aicardi's invariant restricted to n-component links in the solid torus with zero winding number for each component is equal to an invariant we define using the universal cover of the solid torus. We also define and study a geometric invariant for n-component links in the solid torus. We give a lower bound on this invariant using the type one invariants, which are easy to calculate, which helps in computing this geometric invariant, which is usually hard to calculate.     

On region crossing change and incidence matrix

In a recent work of Ayaka Shimizu, she studied an operation named region crossing change on link diagrams, which was proposed by Kishimoto, and showed that a region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that the region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals c n + 1, where c denotes the size of the graph.     

Unsigned State Models for the Jones Polynomial

It is a well-known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in S ³ there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating to the Jones and Bollobás-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.     

The multivariate signed Bollobás-Riordan polynomial

We generalise the signed Bollobás-Riordan polynomial of S. Chmutov and I. Pak [S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial, Mos. Math. J. 7(3) (2007), 409-418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory, Ser. B 99(3) (2009), 617-638. arXiv:0711.3490, doi:10.1016/j.jctb.2008.09.007] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.     

Network modelling and variational Bayesian inference for structure analysis of signed networks

Currently, structure analysis of signed networks with positive and negative links has received wide attention and is becoming a research focus in the area of network science. In recent years, many community detection methods for signed networks have been proposed to analyze the structure of signed networks. However, current methods can only efficiently analyze the signed networks with the single community structure and unable to analyze the signed networks with the coexisting structure of communities and peripheral nodes, bipartite, or other structures. To address this problem, in this study, we present a mathematically principled method for the structure analysis of signed networks with positive and negative links, in which a probabilistic model firstly is proposed to model the signed networks with the single community or the coexisting structure, and a variational Bayesian approach is deduced to learn the approximate distribution of model parameters. For determining the optimal model, we also deduce a model selection criterion based on the evidence theory. In addition, to efficiently analyze the large signed networks, we propose a fast learning version of our algorithm with the time complexity O(k²E) where k is the number of groups and E is the number of links. In our experiments, the proposed method is validated in the synthetic and real-world signed networks, and is compared with the state-of-the-art methods. The experimental results demonstrate that the proposed method can more efficiently and accurately analyze to the structure of signed networks than the state-of-the-art methods.     

Some Calculations on Link Polynomials from 2-Parameter Quantum Groups

Starting with the link invariant G (p,q,z) , a skein relation is introduced which enables us to calculate the polynomial starting with G (p,q,z) = 1, for the unknot. The relation between G (p,q,z) and the known 2-variable link invariant K (l,m) is shown. Then the polynomial of some common knots is calculated, also it is shown that this invariant failed to distinguish the Birmans pairs as well as Jones invariant.     

Khovanov-Jacobsson numbers and invariants of surface-knots derived from Bar-Natan's theory

Khovanov introduced a cohomology theory for oriented classical links whose graded Euler characteristic is the Jones polynomial. Since Khovanov's theory is functorial for link cobordisms between classical links, we obtain an invariant of a surface-knot, called the Khovanov-Jacobsson number, by considering the surface-knot as a link cobordism between empty links. In this paper, we study an extension of the Khovanov-Jacobsson number derived from Bar-Natan's theory, and prove that any -knot has trivial Khovanov-Jacobsson number.

     

Lifting measures to Markov extensions

Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we prove:

1. IfT is anS-unimodal map with an attracting invariant Cantor set, then ∫log|T′|dμ=0 for the unique invariant measure μ on the Cantor set.
2. IfT is piecewise invertible, iff is the Radon-Nikodym derivative ofT with respect to a σ-finite measurem, if logf has bounded distortion underT, and if μ is an ergodicT-invariant measure satisfying a certain lower estimate for its entropy, then μ?m iffh _μ (T)=Σlogf dμ.

     

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

It is known by Loi and Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L^′ to L such that the link L∪L^′ is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian curves.