Invariants Of Plane Curves Research Articles

Vanishing of higher order Alexander‐type invariants of plane curves

AbstractThe higher order degrees are Alexander‐type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves and in terms of the finiteness and the vanishing properties of the invariants of and , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial is a power of , and we characterize when in terms of the defining equations of and . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.

A'Campo-Gusein-Zade diagrams as partially ordered sets

The real analogues of many results about complex monodromies of singularities can be formulated and proved in terms of partial orderings on A'Campo-Gusein-Zade diagrams, the real versions of Coxeter-Dynkin diagrams of singularities. In this paper it is proved that the only diagrams among the A'Campo-Gusein-Zade diagrams of singularities that determine partially ordered sets of finite type (in the sense of representations of a quiver) are the diagrams of simple singularities. To encode the real decompositions of a singularity the analogue of Vasilev invariants turn out to be surjections of a partially ordered set onto a chain. Formulae are proved for Arnold (mod 2)-invariants of plane curves in terms of the corresponding A'Campo-Gusein-Zade diagrams. We define, in the context of higher Bruhat orders, higher partially ordered sets and we describe their connection with the higher M-Morsifications An. We also consider certain previously known results about real singularities from the point of view of partially ordered sets.

Differential 3-knots in 5-space with and without self-intersections

Regular homotopy classes of immersions S 3→ R 5 constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self-intersection. Geometric properties of self-intersections are used to construct two invariants J and St of generic immersions which are analogous to Arnold's invariants of plane curves [1]. We prove that J and St are independent first-order invariants and that any first-order invariant is a linear combination of these. As by-products, some invariants of immersions S 3→ R 4 are obtained. Using them, we find restrictions on the topology of self-intersections.

GENERALIZATION OF ARNOLD'S INVARIANTS OF PLANE CURVES

Considering functions of index at jumps on non-generic curves instead of constants, we define invariants generalizing Arnold's invariants of plane curves. We construct three families of such invariants. We prove that each of these new invariants can be written as a sum of three representatives of these families.

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plucker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J+ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.