We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

Given a link $L$, the Blanchfield pairing $\text{Bl(}L\text{)}$ is a pairing that is defined on the torsion submodule of the Alexander module of $L$. In some particular cases, namely if $L$ is a boundary link or if the Alexander module of $L$ is torsion, $\text{Bl(}L\text{)}$ 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 Blanchfield pairing is Hermitian. Finally, we also obtain short proofs of several properties of $\text{Bl(}L\text{)}$.

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

We consider finitely determined map germs f : (ℝ3, 0) → (ℝ2, 0) with f–1(0) = {0} and we look at the classification of this kind of germ with respect to topological equivalence. By Fukuda's cone structure theorem, the topological type of f can be determined by the topological type of its associated link, which is a stable map from S2 to S1. We define a generalized version of the Reeb graph for stable maps γ : S2→ S1, which turns out to be a complete topological invariant. If f has corank 1, then f can be seen as a stabilization of a function h0: (ℝ2, 0) → (ℝ, 0), and we show that the Reeb graph is the sum of the partial trees of the positive and negative stabilizations of h0. Finally, we apply this to give a complete topological description of all map germs with Boardman symbol Σ2, 1.

The motivation for this work is the state of modern structural mechanisms that are characterized by growing complexity and ever-increasing demands for rapid and accurate motion. These contradictory requirements are often achieved according to easier and easier structures characterized by flexibility segments. In most of cases, the elasticity of structures appears as an obstacle for a precise and rapid control of motion. The aim of this paper is to explore ways of implementation of structural properties of elasticity with the application of high fidelity models during synthesis and analysis of complex mechanisms. Precisely, the aim is to explore the possibility of using Euler–Bernoulli equation, if not in its original form, then to the same extent with the use of modern knowledge in robotics (based on the knowledge of classical mechanics), and to examine the affordability and confirmation of the method through simulation results for a typical robotic configuration.

The paper is concerned with the relationship between the equation of elastic line motion, the “Euler-Bernoulli approach” (EBA), and equation of motion at the point of elastic line tip, the “Lumped-mass approach” (LMA). The Euler–Bernoulli equations (which have for a long time been used in the literature) should be expanded according to the requirements of the motion complexity of elastic robotic systems. The Euler–Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the bending moment of the considered mode. This yields the difference in the structure of Euler–Bernoulli equations for each mode. The stiffness matrix is a full matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral of Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. General form of the elastic line is a direct outcome of the system motion dynamics, and can not be described by one scalar equation but by three equations for position and three equations for orientation of every point on that elastic line. Simulation results are shown for a selected robotic example involving the simultaneous presence of elasticity of the joint and of the link (two modes), as well the environment force dynamics.

We give a necessary condition for a two-bridge knot or link S(p,q) to be 2-adjacent to another two-bridge knot or link S(r,s). In particular, we show that if the trivial knot or link is 2-adjacent to S(p,q), then S(p,q) is trivial, that if S(p,q) is 2-adjacent to its mirror image, then S(p,q) is amphicheiral, and that for a prime integer p, if S(p,q) is 2-adjacent to S(r,s), then S(p,q)=S(r,s) or S(r,s)=S(1,0).

If a hyperbolic link has a prime alternating diagram $D$, then we show that the link complement's volume can be estimated directly from $D$. We define a very elementary invariant of the diagram $D$, its twist number $t(D)$, and show that the volume lies between $v_3(t(D) - 2)/2$ and $v_3(10t(D) - 10)$, where $v_3$ is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite-volume hyperbolic 3-manifolds, in the geometric topology.

This paper gives a number which is used to determine the component number of links from their associated planar graphs. In particular, we use this number to determine the component numbers of links whose associated planar graphs are fans, wheels and 2-sums of graphs.

AMS 2000 Mathematics subject classification: Primary 05C10

We construct a family, $\{HM^{(k)}\}_{k\geq 2}$, of oriented topological equivalence invariants, for projective configurations of complex hyperplanes with fixed combinatorial pattern. We construct a family, $\{PM^{(k)}\}_{k \geq 2}$, of concordance invariants, for classical links with a fixed number of components. The constructions are carried out in the same manner, using the existence of natural additional structures on the nilpotent quotients of the fundamental groups of the complements. We give the explicit computation of the primary and secondary HM and PM invariants.

2000 Mathematical Subject Classification: primary 57M25, 32S50; secondary 20F10, 20F12.

Given a $\text{TQFT}$ in dimension $d\,+\,1$, and an infinite cyclic covering of a closed ($d\,+\,1$)-dimensional manifold $M$, we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a $\text{TQFT}$ and an infinite cyclic covering is then given by the Jordan form of this matrix away from zero. This invariant is also defined if the boundary of $M$ has an ${{S}^{1}}$ factor and the infinite cyclic cover of the boundary is standard. We define a variant of a $\text{TQFT}$ associated to a finite group $G$ which has been studied by Quinn. In this way, we recover a link invariant due to D. Silver and S. Williams. We also obtain a variation on the Silver-Williams invariant, by using the $\text{TQFT}$ associated to $G$ in its unmodified form.

Let A be a finite abelian group and M be a branched cover of an homology 3-sphere, branched over a link L, with covering group A. We show that H1(M; Z[1/|A|]) is determined as a Z[1/|A|][A]-module by the Alexander ideals of L and certain ideal class invariants.

A relation between the Euler characteristics of the Milnor fibres of a real analytic function is derived from a simple identity involving complex monodromy and complex conjugation. A corollary is the result of Coste and Kurdyka that the Euler characteristic of the local link of an irreducible algebraic subset of a real algebraic set is generically constant modulo 4. A similar relation for iterated Milnor fibres of ordered sets of functions is used to define topological invariants of ordered collections of algebraic subsets.

In this paper we prove an inequality that involves the minimal crossing number and the braid index of links by estimating Murasugi and Przytycki’s index for a planar bipartite graph.

We determine the effect on the Jones polynomial evaluated at t = i and t = eπi/3 of an oriented link whenever certain twists are performed.