Victor Goryunov

Lagrangian and Legendrian singularities

These are notes of the introductory courses on the subject we lectured in Trieste in 2003 and Luminy in 2004. The lectures contain basic notions and fundamental theorems of the local theory.

Vassiliev type invariants in arnold's J+-theory of plane curves without direct self-tangencies

Abstract We show that the spaces of complex-valued Vassiliev type invariants for oriented regular plane curves without direct self-tangencies and for oriented framed knots in a solid torus [20] coincide. The isomorphism is provided by the Legendrian lift of plane curves to the solid torus ST * R 2 .

Local invariants of mappings of surfaces into three-space

Following Arnol’s and Viro’s approach to order 1 invariants of curves on surfaces [1, 2, 3, 20], we study invariants of mappings of oriented surfaces into Euclidean 3-space. We show that, besides the numbers of pinch and triple points, there is exactly one integer invariant of such mappings that depends only on local bifurcations of the image. We express this invariant as an integral similar to the integral in Rokhlin’s complex orientation formula for real algebraic curves. As for Arnold’s J + invariant [1, 2, 3], this invariant also appears in the linking number of two legendrian lifts of the image. We discuss a generalization of this linking number to higher dimensions.

Tjurina and Milnor Numbers of Matrix Singularities

To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites f ◦ F with isolated singularities is studied, where f : Y −→C is a function with (possibly non-isolated) singularity and F : X −→Y is a map into the domain of f, and F only is deformed. The corresponding T1(F) is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that τ = μ(f ◦ F) − β0 + β1, where τ is the length of T1(F) and βi is the length of ToriOY(OY/Jf, OX). This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When f has Cohen–Macaulay singular locus (for example when f is the determinant function), relations between τ and the rank of the vanishing homology of the zero locus of f ◦ F are obtained.

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

Abstract. We show that every unframed knot type in

Functions on Space Curves

ST^*{\bf \mathrm{R}}^2

Sectional Singularities and Geometry of Families of Planar Quadratic Forms

has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by V.I.Arnold in [3]. The Legendrian lifting lowers the framed version of the HOMFLY polynomial [20] to generic plane curves. We prove that the induced polynomial invariant can be completely defined in terms of plane curves only. Moreover it is a genuine, not Laurent, polynomial in the framing variable. This provides an estimate on the Bennequin-Tabachnikov number of a Legendrian knot.

Unitary reflection groups and automorphisms of simple hypersurface singularities

We classify simple singularities of functions on space curves. We show that their bifurcation sets have properties very similar to those of functions on smooth manifolds and complete intersections [1,2]: the k(pi, 1)-theorem for the bifurcations diagram of functions is true, and both this diagram and the discriminant are Saitos free divisors.

Kauffman bracket of plane curves

We show that, for hypersurface sections (in the sense of Damon) of isolated functions singularities, the Tjurina and Milnor numbers coincide. An application of this to the families of 2 × 2 symmetric and arbitrary matrices proves the conjectures naturally arising from the results of [2] and [3]. In addition, we study the vanishing homology of the determinantal curves of two-parameter families of symmetric order 2 matrices and construct Dynkin diagrams of simple singularities of such families.

Plane curves, wavefronts and Legendrian knots

In paper [7], generalising Arnold’s approach to boundary singularities, we studied smoothings of simple hypersurfaces invariant under a unitary reflection of finite order. The reflection splits the homology with complex coefficients of a symmetric Milnor fibre into a direct sum of the character subspaces H χ. The monodromy in the space of hypersurfaces with the same symmetry preserves the splitting. It was observed in [7] that the monodromy on each of the H χ is a finite group generated by unitary reflections [13]. This way unitary reflection groups, G(m, 1, k) and seven exceptional groups, made their first appearance in singularity theory addressing one of the long-standing questions of Arnold on realisations of the Shephard-Todd groups [1].