Boundary Links Research Articles

Kirby belts, categorified projectors, and the skein lasagna module of $S^{2} \times S^{2}$

We interpret Manolescu–Neithalath’s cabled Khovanov homology formula for computing Morrison–Walker–Wedrich’s \mathrm{KhR}_{2} skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over Bar-Natan’s cobordism category. Using categorified projectors, we compute the \mathrm{KhR}_{2} skein lasagna modules of (manifold, boundary link) pairs (S^{2} \times B^{2}, \tilde{\beta}) , where \tilde{\beta} is a geometrically essential boundary link, identifying a relationship between the lasagna module and the Rozansky projector appearing in the Rozansky–Willis invariant for nullhomologous links in S^{2} \times S^{1} . As an application, we show that the \mathrm{KhR}_{2} skein lasagna module of S^{2} \times S^{2} is trivial, confirming a conjecture of Manolescu.

Efficient Algorithm for Constructing Order K Voronoi Diagrams in Road Networks

The order k Voronoi diagram (OkVD) is an effective geometric construction to partition the geographical space into a set of Voronoi regions such that all locations within a Voronoi region share the same k nearest points of interest (POIs). Despite the broad applications of OkVD in various geographical analysis, few efficient algorithms have been proposed to construct OkVD in real road networks. This study proposes a novel algorithm consisting of two stages. In the first stage, a new one-to-all k shortest path finding procedure is proposed to efficiently determine the shortest paths to k nearest POIs for each node. In the second stage, a new recursive procedure is introduced to effectively divide boundary links within different Voronoi regions using the hierarchical tessellation property of the OkVD. To demonstrate the applicability of the proposed OkVD construction algorithm, a case study of place-based accessibility evaluation is carried out. Computational experiments are also conducted on five real road networks with different sizes, and results show that the proposed OkVD algorithm performed significantly better than state-of-the-art algorithms.

The frustration-free fully packed loop model

We consider a quantum fully packed loop model on the square lattice with a frustration-free projector Hamiltonian and ring-exchange interactions acting on plaquettes. A boundary Hamiltonian is added to favor domain-wall boundary conditions and link ground state properties to the combinatorics and six-vertex model literature. We discuss how the boundary term fractures the Hilbert space into Krylov subspaces, and we prove that the Hamiltonian is ergodic within each subspace, leading to a series of energy-equidistant exact eigenstates in the lower end of the spectrum. Among them we systematically classify both finitely entangled eigenstates and product eigenstates. Using a recursion relation for enumerating half-plane configurations, we compute numerically the exact entanglement entropy of the ground state, confirming area law scaling. Finally, the spectrum is shown to be gapless in the thermodynamic limit with a trial state constructed by adding a twist to the ground state superposition.

Relative Khovanov–Jacobsson classes

To a smooth, compact, oriented, properly-embedded surface in the $4$-ball, we define an invariant of its boundary-preserving isotopy class from the Khovanov homology of its boundary link. Previous work showed that when the boundary link is empty, this invariant is determined by the genus of the surface. We show that this relative invariant: can obstruct sliceness of knots; detects a pair of slices for $9_{46}$; is not hindered by detecting connected sums with knotted $2$-spheres.

Milnor’s concordance invariants for knots on surfaces

Milnor's $\bar{\mu}$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $\Sigma \times [0,1]$, where $\Sigma$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our extended $\bar{\mu}$-invariants obstruct concordance to homologically trivial knots in thickened surfaces. We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index (or writhe) polynomial, and generalized Alexander polynomial. Furthermore, we complete the slice status classification of all virtual knots up to five classical crossings and reduce to four (out of 92800) the number of virtual knots up to six classical crossings having unknown slice status. Our main application is to Turaev's concordance group $\mathscr{VC}$ of long knots on surfaces. Boden and Nagel proved that the concordance group $\mathscr{C}$ of classical knots in $S^3$ embeds into the center of $\mathscr{VC}$. In contrast to the classical knot concordance group, we show $\mathscr{VC}$ is not abelian; answering a question posed by Turaev.

The mathbb Z-genus of boundary links

The mathbb Z-genus of a link L in S^3 is the minimal genus of a locally flat, embedded, connected surface in D^4 whose boundary is L and with the fundamental group of the complement infinite cyclic. We characterise the mathbb Z-genus of boundary links in terms of their single variable Blanchfield forms, and we present some applications. In particular, we show that a variant of the shake genus of a knot, the mathbb Z-shake genus, equals the mathbb Z-genus of the knot.

Efficient Routing Using Flexible Ethernet in Multi-Layer Multi-Domain Networks

Routing in multi-layer multi-domain (MLMD) networks is challenging due to different technologies and cooperation between different layers and domains. The MLMD routing problem has been considered in prior work, however most of them paid no attention to the inter-layer (or boundary) links, and the inter-domain routing is not yet optimized due to the lack of visibility over the intra-domain network topology. In this article, we investigate the problem of orchestrating MLMD networks by a hierarchical path computation engine (PCE) to leverage the performance of FlexE-the new Flexible Ethernet technology used to link IP and optical domains. We model the routing and FlexE assignment problem for FlexE-Aware and FlexE-Unaware modes and derive an efficient algorithm that optimizes the network utilization by a hierarchical allocation of bandwidth. Facing the issue of missing network information in an MLMD network due to privacy and security reasons, our orchestrator uses a new implicit routing strategy for gathering intra-domain information where the boundary link metrics are considered. Experimental results show that the proposed solution achieves 87% of the optimal throughput, a performance significantly higher than the current practices.

Coevolution of spatial ultimatum game and link weight promotes fairness

In real life, individuals have the capacity to adjust their social relationships on network according to the feedback of their communication. Based on this fact, we present a coevolutionary ultimatum game model to explore the role of dynamic interaction on fairness, in which the link weight is introduced to represent the strength of the social relationship and the link weight is dynamically updated by comparing payoff received via the interaction with the weighted average utility. We mainly analyze the effects of link weight boundary control parameter and link weight adjustment amplitude on the evolution of fairness. Through Monte Carlo simulations, we find that the coevolutionary mechanism of game strategy and link weight can promote the emergence of fairness. More specifically, the fairness is strikingly promoted with the increase of the link weight boundary control parameter, while the influence of link weight adjustment amplitude is not so significant. Moreover, the introduction of this coevolutionary mechanism results in heterogeneous link weight distribution, and the larger link weight boundary control parameter can generate a higher network heterogeneity.

The C-complex clasp number of links

In the 1980s, Daryl Cooper introduced the notion of a C-complex (or clasp-complex) bounded by a link and explained how to compute signatures and polynomial invariants using a C-complex. Since then, this has been extended by works of Cimasoni, Florens, Mellor, Melvin, Conway, Toffoli, Friedl, and others to compute other link invariants. Informally, a C-complex is a union of surfaces which are allowed to intersect each other in clasps. We study the minimal number of clasps amongst all C-complexes bounded by a fixed link L. This measure of complexity is related to the number of crossing changes needed to reduce L to a boundary link. We prove that if L is a 2-component link with nonzero linking number, then the linking number determines the minimal number of clasps amongst all C-complexes. In the case of 3-component links, the triple linking number provides an additional lower bound on the number of clasps in a C-complex.

A family of freely slice good boundary links

We show that every good boundary link with a pair of derivative links on a Seifert surface satisfying a homotopically trivial plus assumption is freely slice. This subsumes all previously known methods for freely slicing good boundary links with two or more components, and provides new freely slice links.

Service environment link and false discovery rate correction: Methodological considerations in population and health facility surveys.

BackgroundGeospatial data are important in monitoring many aspects of healthcare development. Geographically linking health facility data with population data is an important area of public health research. Examining healthcare problems spatially and hierarchically assists with efficient resource allocation and the monitoring and evaluation of service efficacy at different levels. This paper explored methodological issues associated with geographic data linkage, and the spatial and multilevel analyses that could be considered in analysing maternal health service data.MethodsThe 2016 Ethiopia Demographic and Health Survey and the 2014 Ethiopia Service Provision Assessment data were used. Two geographic data linking methods were used to link these two datasets. Administrative boundary link was used to link a sample of health facilities data with population survey data for analysing three areas of maternal health service use. Euclidean buffer link was used for a census of hospitals to analyse caesarean delivery use in Ethiopia. The Global Moran’s I and the Getis-Ord Gi* statistics need to be carried out for identifying hot spots of maternal health service use in ArcGIS software. In addition to this, since the two datasets contain hierarchical data, a multilevel analysis was carried out to identify key determinants of maternal health service use in Ethiopia.ResultsAdministrative boundary link gave more types of health facilities and more maternal health services as compared to the Euclidean buffer link. Administrative boundary link is the method of choice in case of sampled health facilities. However, for a census of health facilities, the Euclidean buffer link is the appropriate choice as this provides cluster level service environment estimates, which the administrative boundary link does not. Applying a False Discovery Rate correction enables the identification of true spatial clusters of maternal health service use.ConclusionsA service environment link minimizes the methodological issues associated with geographic data linkage. A False Discovery Rate correction needs to be used to account for multiple and dependent testing while carrying out local spatial statistics. Examining maternal health service use both spatially and hierarchically has tremendous importance for identifying geographic areas that need special emphasis and for intervention purposes.

Universal surgery problems with trivial Lagrangian

We study the effect of Nielsen moves and their geometric counterparts, handle slides, on good boundary links. A collection of links, universal for 4-dimensional surgery, is shown to admit Seifert surfaces with trivial Lagrangian. They are good boundary links, with Seifert matrices of a more general form than in known constructions of slice links. We show that a certain more restrictive condition on Seifert matrices is sufficient for proving the links are slice. We also give a correction of a Kirby calculus identity in \cite{FK2}, useful for constructing surgery kernels associated to link-slice problems.

An Explicit Computation of the Blanchfield Pairing for Arbitrary Links

AbstractGiven a linkL, the Blanchfield pairing Bl(L) is a pairing that is defined on the torsion submodule of the Alexander module ofL. In some particular cases, namely ifLis a boundary link or if the Alexander module ofLis torsion, Bl(L) can be computed explicitly; however no formula is known in general. In this article, we compute the Blanchfield pairing of any link, generalizing the aforementioned results. As a corollary, we obtain a new proof that the Blanchûeld pairing is Hermitian. Finally, we also obtain short proofs of several properties of Bl(L).

Regenerative Medicine Venturing at the University-Industry Boundary: Implications for Institutions, Entrepreneurs, and Industry.

Regenerative medicine research at university laboratories has outpaced commercial activity. Legal, regulatory, funding, technological, and operational uncertainty have slowed market entry of regenerative medicine treatments. As a result, commercial development has often been led by entrepreneurial ventures rather than large biopharma firms. Translating regenerative medicine across the university-industry boundary links academic scientists, technology transfer organizations, funders, and entrepreneurs. Conflicting motivations among the participants may significantly hinder these efforts. Unproven downstream business models for regenerative medicine delivery further complicate the entrepreneurial process. This chapter explores the challenges associated with entrepreneurial activity commercializing regenerative medicine science developed at research institutions.

On 5–manifolds with free fundamental group and simple boundary links in S5

We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply this to the classification of simple boundary links of $3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed $5$-manifolds with free fundamental group and trivial second homology group.

Rectangular diagrams of surfaces: representability

Introduced here is a simple combinatorial way, which is called a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on and to rectangular diagrams of links. By using rectangular diagrams of surfaces it is intended, in particular, to develop a method to distinguish Legendrian knots. This requires a lot of technical work of which the present paper addresses only the first basic question: which isotopy classes of surfaces can be represented by a rectangular diagram? Roughly speaking, the answer is this: there is no restriction on the isotopy class of the surface, but there is a restriction on the rectangular diagram of the boundary link arising from the presentation of the surface. The result extends to Giroux's convex surfaces for which this restriction on the boundary has a natural meaning. In a subsequent paper, transformations of rectangular diagrams of surfaces will be considered and their properties will be studied. By using the formalism of rectangular diagrams of surfaces an annulus in is produced here that is expected to be a counterexample to the following conjecture: if two Legendrian knots cobound an annulus and have zero Thurston-Bennequin numbers relative to this annulus, then they are Legendrian isotopic. Bibliography: 30 titles.

Principals’ Collaborative Roles as Leaders for Learning

ABSTRACTThis article draws on data from three multicultural New Zealand primary schools to reconceptualize principals’ roles as leaders for learning. In doing so, the writers build on Sinnema and Robinson’s (2012) article on goal setting in principal evaluation. Sinnema and Robinson found that even principals hand-picked for their experience fell short on goal setting. The principals in our study, unsettled by school and classroom conundrums, collaborated with their teachers to deliberately build their TESOL knowledge. We foreground the critical role of TESOL knowledge for principals in setting specific learning goals and valorize distributed leadership that forges non-traditional cross boundary links.

Smooth slice boundary links whose derivative links have nonvanishing Milnor invariants

We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that form a link with nonvanishing Milnor triple linking number. We also give a generalisation to m-component links and higher Milnor invariants. We prove that our examples are ribbon and that all ribbon links are boundary slice.

On the colored Jones polynomials of ribbon links, boundary links and Brunnian links

Habiro gave principal ideals of $\mathbb{Z}[q,q^{-1}]$ in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links

Levels of knotting of spatial handlebodies

If H is a spatial handlebody, i.e. a handlebody embedded in the 3-sphere, a spine of H is a graph Γ ⊂ S3 such that H is a regular neighbourhood of Γ. Usually, H is said to be unknotted if it admits a planar spine. This suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties. Building on this remark, we define several levels of knotting of spatial handlebodies, and we provide a complete description of the relationships between these levels, focusing our attention on the case of genus 2. We also relate the knotting level of a spatial handlebody H to classical topological properties of its complement M = S3 \H, such as its cut number. More precisely, we show that if H is not highly knotted, then M admits special cut systems for M , and we discuss the extent to which the converse implication holds. Along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies. These obstructions are based on recent quandle–coloring invariants for spatial handlebodies, on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links, on the analysis of appropriate coverings of handlebody complements, and on the study of the classical Alexander elementary ideals of their fundamental groups.