Article
The Alexander polynomial as an intersection of two cycles in a symmetric power
We consider a braid β which acts on a punctured plane. Then we construct a local
system on this plane and find a homology cycle D in its symmetric power, such that D · β(D) coincides with the Alexander polynomial of the plait closure of β.
Key words: hyperbolic attractor, generalized solenoid, braid group, mapping class group
The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface S={(z1,z2,z3)∈C3:(z1)5+(z2)3+(z3)2=0} with the 5-dimensional sphere Sε5={(z1,z2,z3)∈C3:|z1|2+|z2|2+|z3|2=ε2}. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S, 0) with the sphere Sε5 of radius ε small enough. Here we discuss to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. We show that, if the strict transform of a curve in (S, 0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding E8-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. (They coincide with each other for a reducible curve singularity and differ by the factor (1−t) for an irreducible one.) We show that, under conditions similar to those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.
An approach by J.Wu describes homotopy groups \pi_n(S^2) of the standard 2-sphere as isotopy classes of spherical n+1--strand Brunnian braids. The case n=3 is investigated for applications.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traffic is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the final node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a finite-dimensional system of differential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of differential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.