Neighborhoods and Manifolds of Immersed Curves

We present some fine properties of immersions ℐ : M ⟶ N between manifolds, with particular attention to the case of immersed curves c : S 1 ⟶ ℝ n . We present new results, as well as known results but with quantitative statements (that may be useful in numerical applications) regarding tubular coordinates, neighborhoods of immersed and freely immersed curve, and local unique representations of nearby such curves, possibly “up to reparameterization.” We present examples and counterexamples to support the significance of these results. Eventually, we provide a complete and detailed proof of a result first stated in a 1991-paper by Cervera, Mascaró, and Michor: the quotient of the freely immersed curves by the action of reparameterization is a smooth (infinite dimensional) manifold.

A relaxed approach for curve matching with elastic metrics

In this paper, we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of H2-metrics with constant coefficients and scale-invariant H2-metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.

HIGHER DEGREE INVARIANTS OF ALMOST GENERIC PLANE IMMERSED CURVES

In earlier works (by Arnold and the author) the approach of singularity theory was used to construct invariants of degree 1 of plane immersed curves. This provides a finer classification of these immersions. In this paper, we construct higher degree invariants by "integrating" some of the basic invariants found earlier of almost generic immersed curves.

STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS

We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.

Invariants of degree one of almost generic plane immersed curves

In 1993, V.I. Arnold used the approach of singularity theory to construct invariants of plane generic immersed curves. This approach suggests a hierarchy of invariants, the coarsest and most fundamental being Arnold's invariants of degree 1. Consider the infinite dimensional space Ω, of all immersions of S1↬ℛ2. The non-generic immersions form a hypersurface called the discriminant which is stratified. The immersions with only one singularity of degree 1 form Σ1, the main part of the discriminant. Given a generic curve on Σ1 (the codim 1 strata), we introduce new invariants of degree 1 in the following sense: when this generic curve passes through Σ2 (immersions with one singularity of degree 2), the value of the invariant jumps by a number which depends only on the stratum of codim 2. The natural stratification of the discriminant yields information about the topology of Σ1, necessary to prove that the invariants are well defined. Of the seen invariants found, five have values in ℤ and two have values in ℤ3. This paper provides an axiomatic description of these invariants.

REPRESENTATIVE BRAIDS FOR LINKS ASSOCIATED TO PLANE IMMERSED CURVES

In [ AC 2], A'Campo associates a link in S3 to any proper generic immersion of a disjoint union of arcs into a 2-disc. We give a sample algorithmic way to produce, from the immersion, a representative braid for such links. As a by-product we get a minimal representative braid for any algebraic link, from a divide associated to a real deformation of the polynomial defining the link.