Classical Link Research Articles

Milnor invariants and twisted Whitney towers

This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Whitney disks. The intersection invariant is a higher-order generalization of the intersection number between two immersed disks in the 4-ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks bounding the given link, together with information that measures the twists (framing obstructions) of the Whitney disks. This interpretation of Milnor invariants as higher-order intersection invariants plays a key role in the classifications of both the framed and twisted Whitney tower filtrations on link concordance (as sketched in this paper). Here we show how to realize the higher-order Arf invariants, which also play a role in the classifications, and derive new geometric characterizations of links with vanishing Milnor invariants of length less than or equal to 2k.

Predicting and Identifying Missing Node Information in Social Networks

In recent years, social networks have surged in popularity. One key aspect of social network research is identifying important missing information that is not explicitly represented in the network, or is not visible to all. To date, this line of research typically focused on finding the connections that are missing between nodes, a challenge typically termed as the link prediction problem . This article introduces the missing node identification problem, where missing members in the social network structure must be identified. In this problem, indications of missing nodes are assumed to exist. Given these indications and a partial network, we must assess which indications originate from the same missing node and determine the full network structure. Toward solving this problem, we present the missing node identification by spectral clustering algorithm (MISC), an approach based on a spectral clustering algorithm, combined with nodes’ pairwise affinity measures that were adopted from link prediction research. We evaluate the performance of our approach in different problem settings and scenarios, using real-life data from Facebook. The results show that our approach has beneficial results and can be effective in solving the missing node identification problem. In addition, this article also presents R-MISC, which uses a sparse matrix representation, efficient algorithms for calculating the nodes’ pairwise affinity, and a proprietary dimension reduction technique to enable scaling the MISC algorithm to large networks of more than 100,000 nodes. Last, we consider problem settings where some of the indications are unknown. Two algorithms are suggested for this problem: speculative MISC, based on MISC, and missing link completion, based on classical link prediction literature. We show that speculative MISC outperforms missing link completion.

Complex trophic interactions of calanoid copepods in the Benguela upwelling system

Life-cycle adaptations, dietary preferences and trophic levels of calanoid copepods from the northern Benguela Current off Namibia were determined via lipid classes, marker fatty acids and stable isotope analyses, respectively. Trophic levels of copepod species were compared to other zooplankton and top consumers. Lipid class analyses revealed that three of the dominant calanoid copepod species stored wax esters, four accumulated triacylglycerols and another three species were characterised by high phospholipid levels. The two biomarker approaches (via fatty acids and stable isotopes) revealed a complex pattern of trophic positions for the various copepod species, but also highlighted the dietary importance of diatoms and dinoflagellates. Calanoides carinatus and Nannocalanus minor occupied the lowest trophic level (predominantly herbivorous) corresponding to high amounts of fatty acid markers for diatoms (e.g. 16:1(n−7)) and dinoflagellates (e.g. 18:4(n−3)). These two copepod species represent the classical link between primary production and higher trophic levels. All other copepods belonged to secondary or even tertiary (some deep-sea copepods) consumers. The calanoid copepod species cover the entire range of δ15N ratios, as compared to δ15N ratios of all non-calanoid taxa investigated, from salps to adult fish. These data emphasise that the trophic roles of calanoid copepods are far more complex than just interlinking primary producers with pelagic fish, which should also be considered in the process of developing realistic food-web models of coastal upwelling systems.

On link encryption against wiretapping attack in network coding

Network coding has been shown to offer an intrinsic security advantage in terms of node-based wiretapping attack. Previous literature explored the security advantage by defining and investigating the security level at an arbitrary intermediate node. As pointed out in other studies and observed by us, in order to achieve the security level at all intermediate nodes, it is necessary to have an admissible link encryption scheme to prevent all links from eavesdropping. “Admissibility” here promises that an adversary cannot obtain more meaningful (decoded or decodable) information about the original message than that disclosed at the node he controls. In this article, we study a similar problem but focus on link encryption. We propose an admissible link encryption scheme LNCLE; in conjunction with LNCLE, linear network coding achieves its security level at all intermediate nodes. In comparison with a trivial link encryption scheme (TLE) which is based on classical link encryption, LNCLE is more efficient in the sense that both key size and computation cost are greatly reduced.

Contact surgery and supporting open books

Let $(M,\xi)$ be a contact 3-manifold. We present two new algorithms, the first of which converts an open book $(\Sigma,\Phi)$ supporting $(M,\xi)$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $(M,\xi)$ into a supporting open book decomposition. These constructions lead to a refinement of a result of Ding-Geiges, which states that every such $(M,\xi)$ may be obtained by contact surgery from $(S^{3},\xi_{std})$, as well as bounds on the support norm and genus of contact manifolds obtained by surgery in terms of classical link data. We then introduce Kirby moves called ribbon moves which use mapping class relations to modify contact surgery diagrams. Any two surgery diagrams of the same contact 3-manifold are related by a sequence of Legendrian isotopies and ribbon moves. As most of our results are computational in nature, a number of examples are analyzed.

Quandle homotopy invariants of knotted surfaces

Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to “regular Alexander quandles”. As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order.

In the Image of Technology: Modern Freedom, Virtual Reality

Liberty was once praiseworthy because it enabled a free choice for the good. Technology recasts the classical link between liberty and virtue by making human well-being depend on technological advance. Till recently, technology signified human mastery over nature. But now virtual reality offers itself as a substitute for nature. Unlike machines which hold us at a distance, it draws us in with images. It no longer shapes the soul indirectly through the effects of using machines, but involves the soul directly. The modern attempt to bracket the soul has brought it back virtually. The argument between fight and flight is at a standstill. Instead, the irony which many lament as a feature of postmodernity can provide a distance from the technologies which have made themselves necessary.

Whitney tower concordance of classical links

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration, and new geometric characterizations of Milnor's link invariants.

GRAPHICAL CALCULI FOR THE DUBROVNIK POLYNOMIAL WITH APPLICATIONS

We introduce a polynomial for plane graphs, which is proposed to equal the Dubrovnik polynomial of the corresponding alternating link diagrams via the medial construction. Then using this polynomial we define another polynomial for doubly edge-weighted plane graphs, which has a natural connection with Dubrovnik polynomial of links formed from plane graphs by edge-tangle replacements. In the final, via this connection we give an explanation for a result due to Lipson and compute the Dubrovnik polynomials of classical pretzel links.

Topological arbiters

This paper initiates the study of topological arbiters, a concept rooted in Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on $RP^2$, which reports the location of the essential 1-cycle. In contrast, we show that there exists an uncountable collection of topological arbiters in dimension 4. Families of arbiters, not induced by homology, are also shown to exist in higher dimensions. The technical ingredients underlying the four dimensional results are secondary obstructions to generalized link-slicing problems. For classical links in the 3-sphere the construction relies on the existence of nilpotent embedding obstructions in dimension 4, reflected in particular by the Milnor group. In higher dimensions novel arbiters are produced using nontrivial squares in stable homotopy theory. The concept of "topological arbiter" derives from percolation and from 4-dimensional surgery. It is not the purpose of this paper to advance either of these subjects, but rather to study the concept for its own sake. However in appendices we give both an application to percolation, and the current understanding of the relationship between arbiters and surgery. An appendix also introduces a more general notion of a multi-arbiter. Properties and applications are discussed, including a construction of non-homological multi-arbiters.

DEALTERNATING NUMBERS AND CLASSICAL LINK INVARIANTS

Dealternating numbers and alternation numbers measure the distance between the link and an alternating links. In the present article, we show that classical link invariants; the determinant, signature and Alexander polynomial can not detect the almost alternativity of links.

Triple linking numbers and triple point numbers of certain [formula omitted]-links

The triple linking number of an oriented surface link was defined as an analogical notion of the linking number of a classical link. We consider a certain m-component T2-link (m⩾3) determined from two commutative pure m-braids a and b. We present the triple linking number of such a T2-link, by using the linking numbers of the closures of a and b. This gives a lower bound of the triple point number. In some cases, we can determine the triple point numbers, each of which is a multiple of four.

RACK SHADOWS AND THEIR INVARIANTS

A rack shadow or rack set is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.

Investigation of classical epidemiological links between patients harbouring identical, non-predominant meticillin-resistant Staphylococcus aureus genotypes and lessons for epidemiological tracking

According to molecular epidemiology theory, two isolates belong to the same chain of transmission if they are similar according to a highly discriminatory molecular typing method. This has been demonstrated in outbreaks, but is rarely studied in endemic situations. Person-to-person transmission cannot be established when isolates of meticillin-resistant Staphylococcus aureus (MRSA) belong to endemically predominant genotypes. By contrast, isolates of infrequent genotypes might be more suitable for epidemiological tracking. The objective of the present study was to determine, in newly identified patients harbouring non-predominant MRSA genotypes, whether putative epidemiological links inferred from molecular typing could replace classical epidemiology in the context of a regional surveillance programme. MRSA genotypes were defined using double-locus sequence typing (DLST) combining clfB and spa genes. A total of 1,268 non-repetitive MRSA isolates recovered between 2005 and 2006 in Western Switzerland were typed: 897 isolates (71%) belonged to four predominant genotypes, 231 (18%) to 55 non-predominant genotypes, and 140 (11%) were unique. Obvious epidemiological links were found in only 106/231 (46%) patients carrying isolates with non-predominant genotypes suggesting that molecular surveillance identified twice as many clusters as those that may have been suspected with classical epidemiological links. However, not all of these molecular clusters represented person-to-person transmission. Thus, molecular typing cannot replace classical epidemiology but is complementary. A prospective surveillance of MRSA genotypes could help to target epidemiological tracking in order to recognise new risk factors in hospital and community settings, or emergence of new epidemic clones.

Application of Dynamic Unstressed State Method in Vertical Rotation Construction of Bridges

A dynamic unstressed length link element and a catenary element are developed based on classical link element and catenary element, and a dynamic unstressed finite element method is proposed which takes unstressed length as input of the finite element and adjusts cable force by modifying unstressed length of element in finite element calculation. Only a finite element model for initial stage of rotation needs to be set in simulating bridge construction of vertical rotation from above to below, and the whole rotation process could be simulated only by adjusting unstressed length of buckle cable and tractive cable, the internal force of buckle, tractive cable and structure in each construction stage can be accurately calculated. This method can also be applied to tension process for cable-stayed bridge, suspension bridges and other cable structure bridges. This paper takes case study to show the correctness of this method and its practical engineering value in application.

Dehn surgeries on some classical links

AbstractWe consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

A NOTE ON THE CROSSING NUMBER AND THE BRAID INDEX FOR VIRTUAL LINKS

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.

THE JONES AND ALEXANDER POLYNOMIALS FOR SINGULAR LINKS

We extend the Kauffman state models of the Jones and Alexander polynomials of classical links to state models of their two-variable extensions in the case of singular links. Moreover, we extend both of them to polynomials with d + 1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.

INDEX POLYNOMIAL INVARIANT OF VIRTUAL LINKS

We introduce a polynomial invariant of virtual links that is non-trivial for many virtuals, but is trivial on classical links. Also this polynomial is sometimes useful to find the virtual crossing number of virtual knots. We give various properties of this polynomial and examples.

A BRACKET POLYNOMIAL FOR GRAPHS, II: LINKS, EULER CIRCUITS AND MARKED GRAPHS

Let D be an oriented classical or virtual link diagram with directed universe [Formula: see text]. Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph [Formula: see text] whose construction involves very little geometric information about the way D is drawn in the plane; consequently [Formula: see text] is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. [Formula: see text] is determined by three things: the structure of [Formula: see text] as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in [Formula: see text] arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of [Formula: see text] is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.