Boundary Links Research Articles

A rational noncommutative invariant of boundary links

In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot. Roughly speaking, this rational presentation of the Kontsevich integral would sum formal power series into rational functions with prescribed denominators. Rozansky's conjecture was soon proven by the second author. We begin our paper by reviewing Rozansky's conjecture and the main ideas that lead to its proof. The natural question of extending this conjecture to links leads to the class of boundary links, and a proof of Rozansky's conjecture in this case. A subtle issue is the fact that a `hair' map which replaces beads by the exponential of hair is not 1-1. This raises the question of whether a rational invariant of boundary links exists in an appropriate space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. A main result of the paper is to construct such an invariant, using the so-called surgery view of boundary links and after developing a formal diagrammatic Gaussian integration. Since our invariant is one of many rational forms of the Kontsevich integral, one may ask if our invariant is in some sense canonical. We prove that this is indeed the case, by axiomatically characterizing our invariant as a universal finite type invariant of boundary links with respect to the null move. Finally, we discuss relations between our rational invariant and homology surgery, and give some applications to low dimensional topology.

A surgery view of boundary links

A celebrated theorem of Kirby identifies the set of closed oriented connected 3-manifolds with the set of framed links in S 3 modulo two moves. We give a similar description for the set of knots (and more generally, boundary links) in homology 3-spheres. As an application, we define a noncommutative version of the Alexander polynomial of a boundary link. Our surgery view of boundary links is a key ingredient in a construction of a rational version of the Kontsevich integral, which is described in subsequent work.

Invariants of boundary link cobordism

Introduction Main results Preliminaries Morita Equivalence Devissage Varieties of representations Generalizing Pfister's theorem Characters Detecting rationality and integrality Representation varieties: Two examples Number theory invariants All division algebras occur Appendix I. Primitive element theorems Appendix II. Hermitian categories Bibliography Index.

ANALYTIC INVARIANTS OF BOUNDARY LINKS

Using basic topology and linear algebra, we define a plethora of invariants of boundary links whose values are power series with noncommuting variables. These turn out to be useful and elementary reformulations of an invariant originally defined by M. Farber [Fa2].

A SURGERY DESCRIPTION OF AN INTEGRAL HOMOLOGY THREE-SPHERE RESPECTING THE CASSON INVARIANT

In 1998, C. Lescop proved that any integral homology sphere with the Casson invariant zero can be obtained from S3 by surgery on a boundary link each component of which has a trivial Alexander polynomial. In this paper, we prove that for any integral homology sphere H, there exists an integer k such that H can be obtained from S3 by surgery on a boundary link each component of which has the Alexander polynomial 1+k(t½-t-½)2.

Calculus of clovers and finite type invariants of 3–manifolds

A clover is a framed trivalent graph with some additional structure, embedded in a 3-manifold. We define surgery on clovers, generalizing surgery on Y-graphs used earlier by the second author to define a new theory of finite-type invariants of 3--manifolds. We give a systematic exposition of a topological calculus of clovers and use it to deduce some important results about the corresponding theory of finite type invariants. In particular, we give a description of the weight systems in terms of uni-trivalent graphs modulo the AS and IHX relations, reminiscent of the similar results for links. We then compare several definitions of finite type invariants of homology spheres (based on surgery on Y--graphs, blinks, algebraically split links, and boundary links) and prove in a self-contained way their equivalence.

A multiresolution texture gradient method for unsupervised segmentation

A texture segmentation algorithm is described, which is designed using a cooperative algorithm within the Multiresolution Fourier Transform (MFT) (Wilson et al., IEEE Trans. Inform. Theory 38(2) (1992) 674–690) framework. The magnitude spectrum of the MFT is employed as feature space wherein the detection of texture boundaries is estimated by means of the combination of boundary information and region properties. A pre-smoothing process is first applied to the MFT magnitudes to reduce the texture fluctuations followed by Sobel operator performed on the MFT magnitudes to give a texture boundary estimate. The refinement of the boundary estimate is accomplished by utilizing both boundary link probabilities and region link probability in an iterative manner. Results on a number of synthetic and natural textures that illustrate the effectiveness of the scheme are presented.

On self Delta-equivalence of boundary links

On self Delta-equivalence of boundary links

Symmetric surgery and boundary link maps

In order to seperate 3-dimensional linking and knotting phenomena, John Milnor introduced the notion of a link homotopy [14]. He allowed selfintersections but did not allow different components to cross during a link homotopy. It is clear that any knot is link homotopically trivial but one of the most surprising and unintuitive (see the left hand side of Fig. 1) results of Milnor was that arbitrarily many parallel copies of a knot form a homotopically trivial link. In fact, one knows that any boundary link has vanishing μ-invariants and thus it is homotopically trivial by the main result of Milnor. By definition, in a boundary link the components bound disjointly embedded Seifert surfaces. For example, any knot has a Seifert surface and is thus a boundary link. Similarly, parallel copies of a knot bound (disjoint) parallel copies of this Seifert surface. The notion of a link homotopy makes clearly sense for links in any dimension. In fact, the correct category to work in seems to be the following: A link map is a continuous map such that connected components in the source are mapped disjointly into the target. A link homotopy between two link maps is then an ordinary homotopy through link maps. In this context, a boundary link map L : qi Mi → N is a link map which is the boundary of a link map qi Wi → N of oriented manifolds Wi, the “Seifert surfaces” for L.

Exploring mutuality within the nurse-patient relationship.

This article explores the concept of boundaries within the nurse-patient relationship. Drawing on child development theories the authors explore the difficulties of boundary maintenance and link this to the professional aspects of therapeutic alliance. The ability to establish clear boundaries between ourselves and others is closely related to the capacity to function as a healthy adult and as such may be said to play an important role in the development of mental health difficulties. However, it is generally accepted by psychological theorists that a certain amount of projection and transference are likely to occur in every two-person relationship and as a result boundary confusion is likely to be experienced by both parties to a greater or lesser extent in the nurse-patient relationship. If, as is currently asserted by the nursing profession, the nurse-patient relationship is based on the principle of equality then it follows that both the nurse and the patient carry equal responsibility for maintaining the boundaries within which their interactions take place. The authors argue that it may be more appropriate to think in terms of a relationship founded on the concept of 'mutuality'. Such a position enables the concept of therapeutic reciprocity to be embraced alongside the reality that ultimately it is the nurse who carries the responsibility for 'holding' the boundaries within which the relationship is 'acted out.'

Realizing homology boundary links with arbitrary patterns

Homology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link. Since a pattern is a set of m m elements which normally generates the free group of rank m m , any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. We will give a constructive geometric proof that all patterns are realized by some homology boundary link L n L^n in S n + 2 S^{n+2} . We shall also prove an analogous existence theorem for calibrations of E \mathbb {E} -links, a more general and less understood class of links tha homology boundary links.

On the kernel of the Casson invariant

We prove: Any integral homology sphere with Casson invariant zero can be obtained from S 3 by surgery on a boundary link each component of which has a trivial Alexander polynomial.

Grain boundary studies in YBCO processed by the liquid phase removal method

Results on grain boundary studies performed on high-current YBCO samples processed by melt texturing with the liquid phase removal method are reported. The misorientation of the a - b planes across each boundary (the kink angle) is measured in several samples. The results indicate that a predominantly large number of high-angle grain boundaries are present in these samples. The high current transport capacity of these bulk polycrystalline samples indicates that the liquid phase removal method is capable of producing good grain boundary links which allow high current to pass through. This lends promise to future applications of bulk YBCO polycrystalline materials

The trivial connection contribution to Witten’s invariant and finite type invariants of rational homology spheres

We derive an analog of the Melvin-Morton bound on the power series expansion of the colored Jones polynomial of algebraically split links and boundary links. This allows us to produce a simple formula for the trivial connection contribution to Witten’s invariant of rational homology spheres. We show that thenth term in the 1/K expansion of the logarithm of this contribution is a finite type invariant of Ohtsuki order 3n and of at most Garoufalidis ordern.

Covering Sato-Levine invariants

Two covering versions of the Sato-Levine invariant are constructed which provide obstructions to certain two-component oriented links in the 3-sphere being link concordant to boundary links. These covering invariants are rational functions one of which detects both nonamphicheirality and noninvertibility of oriented links.

Examining an industrial buyer’s purchasing linkages: a network model and analysis of organizational buying workflow

Employs conceptual contributions from management, work networks, and organizational buying behavior research, and presents the results of a study which integrates these contributions through an internal marketing exchange model. Specifically, network analysis is used to describe purchasing workflow patterns within a single industrial firm. Both prescribed networks (i.e. hierarchical level departmental membership and product purchasing assignment) and emergent networks (i.e. position on the organizational boundary and centrality links within the firm’s buying system) are investigated. In addition, the position of a particular buyer is discussed in terms of his positional role within the organization’s internal marketing exchange system. Results provide implications for purchasing managers and organizational buying researchers, and directions for future research are discussed.

Homology boundary links and Blanchfield forms: concordance classification and new tangle-theoretic constructions

THIS work forms part of our on-going effort to classify the set of concordance classes of links.RecallthatalinkL= (K,,. . . ,K,}inS “+’ is a locally flat piecewise-linear, oriented submanifold of S”+2 of which each component Ki is homeomorphic to S”. The exterior E(L) of a link L is the closure of the complement of a small regular neighborhood N(L) of L. A longitude of a component Ki is a parallel of Ki lying on the boundary of the tubular neighborhood (untwisted if n = 1). A meridian pi is a path from a basepoint to JN(L) which traverses a fiber of 8N(L) and returns. A Seifert Surface for Ki is a connected, compact, oriented, (n + 1)-manifold K E E(L) such that aK is a longitude of Ki. Links Lo, L1 are concordant (or cobordant) if there is a smooth, oriented submanifold C = {C,, . . . , C,} of S n+2 x [0, l] which meets the boundary transversely in X, is piecewise-linearly homeomorphic to L,, x [0, 11, and meets Sn+2 x {i} in Li for i = 0, 1. In the mid-603 M. Kervaire and J. Levine gave an algebraic classification of knot concordance groups (m = 1) in high dimensions (n > 1) [l]. For even n these are trivial and for odd n they are infinitely generated, being isomorphic to certain Witt groups obtained from information garnered from the Seifert surface. Extending Levine’s knot cobordism classification to links is difficult for several reasons. Firstly, if m > 1, the natural operation of connected-sum is not well-defined on concordance classes so there is no obvious group structure. Secondly, the Seifert surfaces for different components of a link may intersect, obstructing at least the naive generalization of the Seifert form information. However, the techniques do extend well to the class of boundary links. A boundary link is one which admits a collection of m disjoint Seifert surfaces, one for each component. In fact, S. Cappell and J. Shaneson classified boundary links modulo boundary link cobordism in 1980 using their homology surgery groups, followed later by Ki Ko and W. Mio who accomplished this via Seifert surfaces [2-41. A boundary link cobordism is a cobordism C between Lo and L1 for which there exist disjointly embedded 2n-manifolds IV= {IV,, . . . , ZVm) in E(C) such that IVn(S”+2 x {i}) is a system of Seifert surfaces for the boundary link Li, i = 0, 1, and such that

Correction to : "A boundary link is trivial if the Lusternik-Schnirelmann category of its complement is one''

Correction to : "A boundary link is trivial if the Lusternik-Schnirelmann category of its complement is one''

Classical knot and link concordance

In [G1], we combined the slicing obstructions of Levine [L] with those of Casson and Gordon [CGI], [CG2], [Go] in a nontrivial way. Essentially we related the metabolizer for the Seifert form of a slice knot which Levine guaranteed to the characters on the homology of the 2-fold branched cover (with vanishing Casson Gordon invariants) which Casson and Gordon guaranteed. In this paper we will simultaneously consider all the q-fold branch covers for q a prime power. We will define F + which strengthens the group F ' of [GI]. In [GL2] Livingston and Gilmer applied these methods to concordances of two component links with linking number zero. They defined an algebraic group ~ which enhanced F' to study concordances of links with two components. In this paper we will define an analogous strengthening ~ ' of ~, We also calculate enough Casson-Gordon invariants to decide when a genus one knot maps to zero in F +. Finally we give a new example of a link which is a fusion of a boundary link but not concordant to a boundary link. Cochran and Orr [CO] were the first to discover such links. Livingston was the first to observe that our work on Casson Gordon invariants could be applied to this problem [Li]. We would like to thank Chuck Livingston for many valuable conversations. We work in the smooth category.

Not all Links are Concordant to Boundary Links

Not all Links are Concordant to Boundary Links